optimal pricing strategies and customers’ equilibrium...

Research ArticleOptimal Pricing Strategies and Customersrsquo EquilibriumBehavior in an Unobservable MM1 Queueing System withNegative Customers and Repair

Doo Ho Lee

Department of Industrial and Management Engineering Kangwon National University 346 Joongang-roSamecheok-si Gangwon-do 29513 Republic of Korea

Correspondence should be addressed to Doo Ho Lee enjdhleegmailcom

Received 15 June 2017 Revised 29 September 2017 Accepted 15 October 2017 Published 8 November 2017

Academic Editor Vladimir Turetsky

Copyright copy 2017 Doo Ho Lee This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This work investigates the optimal pricing strategies of a server and the equilibrium behavior of customers in an unobservableMM1 queueing system with negative customers and repair In this work we consider two pricing schemes The first is termedthe ex-post payment scheme where the server charges a price that is proportional to the time spent by a customer in the systemThe second scheme is the ex-ante payment scheme where the server charges a flat rate for all services Based on the reward-coststructure the server (or system manager) should make optimal pricing decisions in order to maximize its expected profit per timeunit in each payment scheme This study also investigates equilibrium joiningbalking behavior under the serverrsquos optimal pricingstrategies in the two pricing schemes We show given a customerrsquos equilibrium that the two pricing schemes are perfectly identicalfrom an economic point of view Finally we illustrate the effect of several system parameters on the optimal joining probabilitiesthe optimal price and the equilibrium behavior via numerical examples

1 Introduction

Over the past several decades queueing systems consid-ering customersrsquo equilibrium behavior have received muchattention owing to their applications to the management ofvarious systems This type of economic analysis of queueingsystems was pioneered by Naor [1] who was the first tostudy an observable single-server queueing system Naor [1]introduced the equilibrium and social optimal strategies forthe joining problem in an MM1 queue with a simple linearreward-cost structure Naorrsquos model was extended by severalresearchers including Yechiali [2] Johansen and Stidham [3]Stidham [4] andMendelson andWhang [5] Chen and Frank[6] studied Naorrsquos model while assuming that both the cus-tomers and the server maximize their expected discountedutility using a common discount rate Larsen [7] investigatedanother generalization of Naorrsquos model assuming that thecustomers differ according to their service values Erlichmanand Hassin [8] analyzed a single-server Markovian queue

which allowed customers to overtake others Edelson andHildebrand [9] studied an unobservable case in which cus-tomers make decisions without information about the stateof the system Hassin and Haviv [10] conducted an economicanalysis of a priority queue where two priority classes canbe purchased and they derived Nash equilibrium strategies(pure or mixed) of the threshold type In contrast to classicalqueueing systems customers are allowed to make decisionsbased on the information available upon their arrival in thesesystems Therefore the systems can be modeled as gamesamong arriving customers and the fundamental problem isto identify the Nash equilibrium The monograph of Hassinand Haviv [11] summarized the main approaches and severalresults with various observable and unobservable queueingsystems offering extensive bibliographical references

Since the concept was introduced by Gelenbe [12] therehas been a rapid increase in the amount of literature onqueueing systems with negative customers due to the appli-cation to neural networks communication systems and

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 8910819 11 pageshttpsdoiorg10115520178910819

2 Mathematical Problems in Engineering

servicemanufacturing systems If a negative customer arrivesat a system it removes one ordinary customer (called apositive customer) according to a predetermined removaldiscipline and the server breaks down Because the perfor-mance measure of such a systemmay be influenced by serverfailure and repair operations such an unreliable system iswell worth analyzing in light of both queueing theory andreliability theory Additional studies on unreliable queueingsystems were conducted by Gelenbe [13] Artalejo [14] Chaeet al [15] and Lee et al [16] and references therein

However few works have attempted an equilibriumanalysis of an unreliable queueing system despite the factthat server failure is a common phenomenon in everydaylife For example equilibrium strategies with various levelsof information were studied by Economou and Kanta [17]Boudali and Economou [18 19] and Economou and Manou[20] considered equilibrium balking strategies in an unre-liable single-server queue with catastrophes (or stochasticclearing) where the server fails and all present customersin the system are forced to leave upon the catastrophersquosarrival Zhang et al [21] analyzed optimal and equilibriumretrial rates in an unreliable queue in which the server issubject to vacations along with breakdowns and repair Wanget al [22] studied the equilibrium behavior of customersin a retrial queue with negative customers and derivedcorresponding Nash equilibrium joining strategies and asocial net benefit maximization problem with respect to thelevels of information available to customers upon arrival Liet al [23] conducted an equilibrium analysis on unreliableMM1 queueing systems with working breakdowns Whenthe system is under repair the secondary (substitute) serverprovides services to customers at a slow speed instead of stop-ping servicing completelyWang andZhang [24] extended themodel in [17] by assuming that the failed server experiencesan exponential delay before the repair begins Yu et al [25]also dealt with the same model of [24] for an unobservablecase

Veltman and Hassin [26] presented an interesting paperon the pricing strategies of a server in a Markovian MM1queueing system They analyzed equilibrium pricing strate-gies under two types of pricing structures In the first pricingstructure the server charges a fee that is proportional tothe sojourn time (the waiting time plus the service time)referred to as an ex-post payment (EPP) scheme In thesecond however the server charges a fixed fee for theservice meaning that the server implements an ex-ante pay-ment (EAP) scheme Under each pricing scheme the equi-librium joining behavior of customers was examined andcompared with socially optimal results Later Sun et al [27]extended this analysis to a batch arrival queueing system andEconomou and Kanta [28]Wang and Zhang [29] and Zhanget al [30] studied retrial cases Recently Ma and Liu [31]considered pricing strategies in a discrete-time GeoGeo1queueing system and Lee [32] studied the optimal pricingstrategies in a discrete-timeGeoGeo1 queuing systemunderthe EPP and EAP schemes by assuming the sojourn time-dependent reward

A queueing system with negative customers can be foundin many real-life systems One example is a cloud data center

where the scheduler allocates a virtual machine (VM) fora service request which is generated by a user All arrivingservice requests are initially buffered in the scheduler queueafter which each is transmitted to a certain VM according toa predetermined scheduling policy built into the schedulerIn addition the scheduler oversees service request processingand the VM status for the billing of services Dependingon the operating conditions of the cloud data center thescheduler charges a fixed price regardless of the system usagetime or charges a price that is proportional to the timeTherefore the scheduler plays a key role in operating thecloud data center stably andmaximizing revenue Because thescheduler is the single point of failure of the entire cloud datacenter a secondary scheduler must be built in preparation fora system failure network hacking or high availability Oncethemain scheduler fails the request in service is deleted fromthe scheduler and the secondary scheduler replaces the mainscheduler While the main scheduler is being repaired thesecondary scheduler stores the arriving service requests onlyin the scheduler queueTherefore in terms ofmaximizing theoperation profits of the cloud data center analyzing optimaland equilibrium pricing strategies is important

Motivated by the aforementioned studies and the possibleapplication to pricing in a cloud data center we analyze opti-mal pricing strategies and the equilibrium joiningbalkingbehavior of customers in an unobservable MM1 queue-ing system with negative customers and repair under thetwo pricing schemes described above Our contributionsare twofold First we derive not only the optimal joiningstrategies of customers but also the optimal prices set bythe server in equilibrium under our model To the authorrsquosknowledge this study is the first to consider the pricingstrategies together with the customer equilibrium behaviorfor an MM1 queueing system with server failure and repairSecond under the EPP and EAP schemes we show thatthe two pricing schemes do not affect customersrsquo joiningbehavior and the serverrsquos maximum profits which has notbeen reported in the literature on anMM1 queueing systemwith negative customers and repair

This paper is structured as follows In Section 2 wedescribe the model and derive the expected sojourn timeof an unobservable MM1 queueing system with negativecustomers and repair In Sections 3 and 4 customersrsquo equi-librium joining behavior and the serverrsquos profit maximizationstrategies under the two pricing schemes are driven in a par-tially unobservable case and in a fully unobservable case Wealso deal with numerical experiments in which we investigatetrends in the optimal prices and joining probabilities of EPPand EAP schemes according to various input values

2 Preliminaries

This work considers a queueing system with the followingfeatures Ordinary customers arrive at a single server queueaccording to a Poisson process with a rate of 120582 (120582 gt 0)A server renders a service to each ordinary customer on afirst-come first-out basis The service time is exponentiallydistributed with a rate of 120583 (120583 gt 0) Negative customersarrive only when the server is in operation according to a

Mathematical Problems in Engineering 3

Poisson process with a rate of 120578 (120578 gt 0) We assume thefollowing removal discipline each time a negative customerarrives while the server is in operation the server breaksdown and the ordinary customer being served is forced toleave the system If the negative customer arrives at an emptysystem it simply triggers a server failure because there is nocustomer who can leave When the server fails the server isturned off and a repair process begins immediatelyThe repairtime for the failed server is exponentially distributed at a rateof 120574 (120574 gt 0) While the server is under repair the stream ofthe arrival process of ordinary customers continues

Remark 1 In many cases a Poisson distribution can modelthe number of occurrences of rare events such as the arrival ofrequests the service process and service failures in informa-tion communication systems In a cloud data center while thenumber of potential users is high each user typically submitsa task at any given time with a low probability Thereforeservice requests can be adequately modeled as a Poissonprocess Moreover since a service failure is not a frequentlyobserved event we can assume that negative customers arriveaccording to a Poisson process In the context of cloud datacenter applications however it is difficult to conclude that theservice process or repair process follows a Poisson processdue to the synchronized communication of packets in datacenters Nevertheless because one of the objectives of ourwork is to investigate the joining behavior of customersand related pricing strategies we assume that the servicetime and repair time follow the exponential distribution foranalytical simplicity As shown in the literature on cloud datacenter applications Markovian queueing systems have beenused to model actual system behavior and to determine theperformance measures of systems [33 34]

Whenever an ordinary customer arrives at the systemheshe decides to join or balk depending on the specificjoining probability An arriving ordinary customer joins thequeue with a probability of 1199020 when the server is under repairOtherwise heshe does with a probability of 1199021 when theserver is in operation

Let 119873(119905) be the number of (ordinary) customers in thesystem at time 119905 with the following definitions

119868 (119905) = 1 The server is in operation at time 1199050 The server is under repair at time 119905 (1)

The vector process (119873(119905) 119868(119905)) 119905 ge 0 then becomes a two-dimensional continuous-time Markov chain with the statespace expressed as Ω = (119899 0) 119899 ge 0 cup (119899 1) 119899 ge 0and its nonzero transition rates are given by119903(119899119894)(119899+1119894) = 120582119902119894 119899 ge 0 119894 = 0 1119903(1198991)(119899minus11) = 120583 119899 ge 1119903(01)(00) = 120578119903(119899+11)(1198990) = 120578 119899 ge 0119903(1198990)(1198991) = 120574 119899 ge 0

(2)

The corresponding transition diagram is shown in Figure 1Define 119901(119899 119894) = lim119905rarrinfin Pr119873(119905) = 119899 119868(119905) = 119894 (119899 119894) isin Ω

The balance equations governing the system are then given by(1205821199020 + 120574) 119901 (0 0) = 120578119901 (0 1) + 120578119901 (1 1) (3)(1205821199020 + 120574) 119901 (119899 0) = 120578119901 (119899 + 1 1)+ 1205821199020119901 (119899 minus 1 0) 119899 ge 1 (4)

(1205821199021 + 120578) 119901 (0 1) = 120574119901 (0 0) + 120583119901 (1 1) (5)(1205821199021 + 120583 + 120578) 119901 (119899 1) = 120574119901 (119899 0) + 1205821199021119901 (119899 minus 1 1)+ 120583119901 (119899 + 1 1) 119899 ge 1 (6)

From (3)ndash(6) we now have the following lemma

Lemma 2 Define119882(1199020 1199021) as the expected sojourn time of anordinary customer who decides to join the queue upon hisherarrival After some calculations119882(1199020 1199021) can be obtained as

119882(1199020 1199021) = 1205741199021 (1205821205781199021 + 120574 (120583 + 120578)) + 1205781199020 ((120583 + 120578) (120583 + 120574 + 120578) minus 1205821199021 (120583 + 120574)) + 12058212058312057811990220(120583 + 120578) (1205781199020 + 1205741199021) (120574 (120583 + 120578) minus 120582 (1205781199020 + 1205741199021)) (7)

where 120582(1205781199020+1205741199021) lt 120574(120583+120578) should hold for the stable system

Proof See Appendix A

We assume that an ordinary customer receives a rewardof 119877 (119877 gt 0) units for departing the system Moreover hesheaccumulates costs at a rate of 119862 (119862 gt 0) units per unittime during which heshe remains in the system Ordinarycustomers are risk-neutral and their decisions are irrevocablein the sense that retrials of balking customers and renegingof entering customers are not allowed To make the modelnontrivial we assume a condition that 119877 gt 119862((120583+120578)minus1+120574minus1)

which ensures that the reward for joining the queue exceedsthe expected cost of an ordinary customer finding the systemstate (0 0)Remark 3 Under our assumption it may not sound plausiblethat a customer whose service due to a server failure iscanceled receives the same reward of 119877 The author agreesin fact Boudali and Economou [18 19] assumed that eachcustomer receives either a reward of 119877119904 units for completingthe service or failure compensation of 119877119891 units if forced toabandon the system due to a server failure However thispaper assumes that all joining customers receive the same


0 0

0 1

1 0

1 1

2 0 n 0

2 1 n 1

q1q1q1q1q1

q0q0q0q0q0

middot middot middotmiddot middot middot

middot middot middot middot middot middot

Figure 1 Transition diagram for the partially unobservable case

reward for simplicity of the model and the analysis Themain purposes of this paper are to compare customer joiningbehavior in the EPP and EAP schemes and to determinethe optimal price of each scheme under a linear reward-cost structure We will leave the model in which customersreceive different amounts of compensation depending on thecompletion of service as future study

We distinguish the following two cases based on theinformation available to ordinary customers upon theirarrival (i) the partially unobservable case where ordinarycustomers observe the server state 119868(119905) but not the number ofcustomers in the system119873(119905) and (ii) the fully unobservablecase where ordinary customers are not informed about 119868(119905)or119873(119905)3 Pricing Analyses in the PartiallyUnobservable Case

In this section we investigate a partially unobservable casewhere arriving ordinary customers observe the state of theserver upon their arrival but not the number of customersin the system There are four pure strategies available for acustomer to join the queue when the server is under repairor in operation or not to join the queue when the server isunder repair or in operation An ordinary customerrsquos joiningstrategy can be described by the two probabilities of 1199020 and1199021 (0 le 119902119894 le 1) which are the joining probabilities

Remark 4 In fact the model under our study can beviewed as an MM1 queue under a random environmentSpecifically the external environment 119868(119905) is an irreduciblecontinuous-time Markov chain in the finite state space 0 1The infinitesimal generator of the external environment isgiven by

Q = (minus120574 120574120578 minus120578) (8)

With (1205870 1205871) as the stationary distribution of the externalenvironment after solving (1205870 1205871)Q = (0 0) we have 1205870 =120578(120574 + 120578)minus1 and 1205871 = 120574(120574 + 120578)minus1 That is 1205870 (1205871) representsthe stationary probability that the server is under repair (inoperation)Therefore the effective arrival rate 120582eff is equal to12058211990201205870 + 12058211990211205871 = 120582(1205781199020 + 1205741199021)(120574 + 120578)minus1 due to the PASTAproperty [35]

31 EPP Scheme Under the EPP scheme the server chargesthe price in proportion to the customerrsquos sojourn time Let119870119905 (119870119905 ge 0) and 119875119905 denote the price charged by the server(or system) and the expected server profit per time unitrespectively The expected benefit 119880119905 then has the followingrelationship 119880119905 = 119877 minus (119870119905 + 119862)119882(1199020 1199021) If 119880119905 equals zero atcustomerrsquos equilibrium we have 119877 = (119870119905 + 119862)119882(1199020 1199021) Atcustomerrsquos equilibrium therefore the price can be expressedin terms of the customerrsquos joining probabilities denoted by1199020119905 and 1199021119905 119870119905 = 119877119882(1199020119905 1199021119905) minus 119862 (9)

Substituting (9) into 119875119905 = 120582eff119870119905119882(1199020119905 1199021119905) we have119875119905 = 120582 (1199020119905120578 + 1199021119905120574)119870119905119882(1199020119905 1199021119905)120574 + 120578= 120582 (1199020119905120578 + 1199021119905120574) (119877 minus 119862119882(1199020119905 1199021119905))120574 + 120578 (10)

We establish the following nonlinear programming(NLP) problem to maximize 119875119905 with respect to 1199020119905 and 1199021119905

max1199020119905 1199021119905

119875119905 = 120582 (1199020119905120578 + 1199021119905120574) (119877 minus 119862119882(1199020119905 1199021119905))120574 + 120578st 0 le 119902119894119905 le 1 119894 = 0 1120582 (1205781199020119905 + 1205741199021119905) lt 120574 (120583 + 120578)

(11)

In (11) we want to maximize the expected server profit pertime unit under the EPP scheme in the partially unobservablecase The first constraint implies that the joining probabilityshould be bounded between 0 and 1 and the second guar-antees the system to be stable We introduce the followinglemma

Lemma 5 The optimization problem in (11) is a convexmaximization problem (CMP)

Proof See Appendix B

From Lemma 5 we now have the following theorem

Theorem 6 Under the EPP scheme in the partially unobserv-able case if 0 le 1199020119905 le 1 and 0 le 1199021119905 le 1 there exists a uniqueequilibrium where customers join the queue with probabilitiesof 119902lowast0119905 and 119902lowast1119905 the optimal price which maximizes the expectedserver profit per unit time is given by119870lowast119905 = 119877119882(119902lowast0119905 119902lowast1119905) minus119862Proof Because the optimization problem in (11) is a CMP theLagrange multiplier method can be used to find the optimalsolution Observing (11) we find that all constraints areinequality constraints thus Karush-Kuhn-Tucker conditionscan be used to generalize themethod of LagrangemultipliersHowever the explicit formof the optimal 119902119894119905 119894 = 0 1 denotedby 119902lowast119894119905 is too long and complicated Instead we introduceanother means by which the value of 119902lowast119894119905 can be obtained


According to Boyd and Vandenberghe [36] if the objectivefunction is strictly concave in the feasible region a uniquelocal optimal solution obtained through the Newton methodbecomes a unique global optimal solution Details about theNewton method can be found in Boyd and Vandenberghe[36] This completes the proof

One may raise the question of how we can say that 119902lowast0119905and 119902lowast1119905 are also equilibrium joining probabilities as they areaffected by the price charged by the server As mentionedearlier the expected benefit119880119905 has the following relationship119880119905 = 119877 minus (119870lowast119905 + 119862)119882(119902lowast0119905 119902lowast1119905) We distinguish the followingthree cases

Case 1 (119877 le (119870lowast119905 + 119862)119882(0 0)) In this case even if noother customer joins the expected benefit of a customer whojoins is nonpositive Therefore the strategy of joining withprobabilities 119902lowast0119905 = 119902lowast1119905 = 0 is an equilibrium strategy andno other equilibrium is possible Moreover in this case notjoining is the dominant strategy

Case 2 (119877 ge (119870lowast119905 + 119862)119882(1 1)) In this case even if allpotential customers join they all enjoy a nonnegative benefitTherefore the strategy of joining with probability 119902lowast0119905 =119902lowast1119905 = 1 is an equilibrium strategy and no other equilibriumis possible Moreover in this case joining is the dominantstrategy

Case 3 ((119870lowast119905 + 119862)119882(0 0) lt 119877 lt (119870lowast119905 + 119862)119882(1 1)) Inthis case if 119902lowast0119905 = 119902lowast1119905 = 1 then a customer who joinssuffers a negative benefit Hence this cannot be an equi-librium strategy Likewise if 119902lowast0119905 = 119902lowast1119905 = 0 a customerwho joins receives a positive benefit more than by balkingHence this cannot be an equilibrium strategy Thereforethere exists unique equilibrium joining probabilities 119902lowast0119905 and119902lowast1119905 satisfying 119877 = (119870lowast119905 + 119862)119882(119902lowast0119905 119902lowast1119905) for which customersare indifferent with regard to joining andor balking Assumethat 119877 = 20 119862 = 2 120582 = 1 120583 = 08 120574 = 15 and 120578 = 005FromTheorem 6 we obtain the optimal joining probabilities119902lowast0119905 = 034801 and 119902lowast1119905 = 054417 and the optimal price119870lowast119905 = 378749 Therefore 119882(119902lowast0119905 119902lowast1119905) = 345577 and therelationship 119877 = (119870lowast119905 + 119862)119882(119902lowast0119905 119902lowast1119905) should hold

32 EAP Scheme Model Next We consider the EAP schememodel where the server charges a flat rate for services Let119870119891 and 119875119891 be the fixed price charged by the server and theserverrsquos expected profit per time unit respectively Given thatthe expected benefit for a customer is expressed as 119880119891 = 119877 minus119870119891 minus 119862119882(1199020 1199021) and equals zero in customersrsquo equilibriumwe obtain 119877 = 119870119891 + 119862119882(1199020 1199021) Let 1199020119891 and 1199021119891 denote thecustomerrsquos joining probabilities in the EAP scheme 119870119891 is afunction of 1199020119891 and 1199021119891 and has the form of119870119891 = 119877 minus 119862119882(1199020119891 1199021119891) (12)

Substituting (12) into 119875119891 = 120582eff119870119891 we have119875119891 = 120582119870119891 (1199020119891120578 + 1199021119891120574)120574 + 120578

= 120582 (1199020119891120578 + 1199021119891120574) (119877 minus 119862119882(1199020119891 1199021119891))120574 + 120578 (13)

We establish the following NLP problem to maximize 119875119891with respect to 1199020119891 and 1199021119891

max1199020119891 1199021119891

119875119891= 120582 (1199020119891120578 + 1199021119891120574) (119877 minus 119862119882(1199020119891 1199021119891))120574 + 120578

st 0 le 119902119894119891 le 1 119894 = 0 1120582 (1205781199020119891 + 1205741199021119891) lt 120574 (120583 + 120578) (14)

In (14) we also want to maximize the expected server profitper time unit under the EAP scheme in the partially unob-servable case The first constraint implies that the joiningprobabilities should be bounded between 0 and 1 and thesecond specifies the stability condition It should be notedthat the optimization problem in (14) is perfectly identicalto the CMP in (11) In other words the optimal joiningstrategies for the customer and themaximumexpected serverprofits are identical in the two different pricing schemes Weintroduce the following theorem

Theorem 7 Under the EAP scheme in the partially unobserv-able case if 0 le 1199020119891 le 1 and 0 le 1199021119891 le 1 there exists a uniqueequilibrium where customers join the queue with probability119902lowast119894119891 = 119902lowast119894119905 119894 = 0 1 and the optimal price which maximizesthe expected server profit per unit time is determined by 119870lowast119891 =119877 minus 119862119882(119902lowast0119891 119902lowast1119891)Proof From Lemma 8 the CMP in (14) is perfectly identicalto that in (11) which means that 119902lowast119894119891 = 119902lowast119894119905 119894 = 0 1 Theoptimal price whichmaximizes the expected server profit perunit time is calculated by inserting 119902lowast0119891 and 119902lowast1119891 into (12)Thiscompletes the proof

33 Numerical Experiments in the Partially UnobservableCase Based on the results of the partially unobservablecase we compare the joining probability of the EPP schemeand that of the EAP scheme with various experimentalparameters We set the common experimental parameters as(119877 119862) = (20 2)

Figure 2(a) shows the trends of the optimal joiningprobabilities 119902lowast0 and 119902lowast1 according to the service rate 120583 Notethat 119902lowast119894 = 119902lowast119894119905 = 119902lowast119894119891 119894 = 0 1 We confirm that 119902lowast0 and119902lowast1 are increasing functions of 120583 It is clear that when theserver renders faster service more customers are likely tojoin the queue We also observe that 119902lowast0 lt 119902lowast1 If an arrivingcustomer joins the queue while the server is under repairheshe should wait the additional remaining repair timeThishas a negative impact on the customersrsquo joining behaviorTherefore customers prefer to join the queue when the serveris in operation


04

05

06

07

08

Opt

imal

join

ing

prob

abili

ty

09 10 11 1208Service rate ()

qlowast1

qlowast0

(a) 120582 = 1 120574 = 15 and 120578 = 005

02

04

06

08

10

Opt

imal

join

ing

prob

abili

ty

10 15 20 25 30 35 4005Repair rate ()

qlowast1

qlowast0

(b) 120582 = 1 120583 = 1 and 120578 = 05

02

04

06

08

10

Opt

imal

join

ing

prob

abili

ty

05 10 15 20 25 3000Negative customer arrival rate ()

qlowast1

qlowast0

(c) 120582 = 1 120583 = 1 and 120574 = 1

02

04

06

08

Opt

imal

join

ing

prob

abili

ty

09 10 11 1208Ordinary customer arrival rate ()

qlowast1

qlowast0

(d) 120583 = 1 120574 = 1 and 120578 = 005

Figure 2 Optimal joining probabilities

Figure 2(b) shows the trends of 119902lowast0 and 119902lowast1 according tothe repair rate 120574 We observe that 119902lowast0 and 119902lowast1 are increasingfunctions of 120574 As the repair speed increases the probabilitythat the server is in operation also increases Thereforemore customers are likely to join the queue and receiveservices due to the high reliability of the server Moreoveras 120574 increases 119902lowast0 increases more sharply increasing than119902lowast1 That is the repair rate has a stronger positive impact oncustomers arriving when the server is under repair than onthose arriving when the server is in operation

Figure 2(c) shows the trends of 119902lowast0 and 119902lowast1 according to thenegative customer arrival rate 120578 The interesting observationin Figure 4 is that 119902lowast0 sharply increases after 119902lowast1 reaches 1 Thisphenomenon that is that customers who arrive while theserver is under repair tend to imitate the behavior of thosewho arrive while the server is in operation is known as theldquoFollow-The-Crowdrdquo behavior

Figure 2(d) shows that both 119902lowast0 and 119902lowast1 are decreasingfunctions of the ordinary customer arrival rate 120582 As morecustomers arrive the congestion of the system will increaseimplying that higher (external) arrival rate leads to a lowerprobability of actual joining

4 Pricing Analyses for the FullyUnobservable Case

In this section we consider the fully unobservable case wherean arriving ordinary customer cannot observe the systemstate at all There are two pure strategies available for acustomer to join or not to join An ordinary customerrsquosjoining strategy can be described by a probability of 119902 (0 le119902 le 1) which is the probability of joining and the effectivearrival rate (joining rate) is 120582119902The transition diagram for thiscase is illustrated in Figure 3

In the fully observable case all customers have the samejoining probability of 119902 therefore by considering 1199020 = 1199021 = 119902in (7) we have

119882(119902) = 120574 (120574 + 120578) + 120578 (120583 + 120578)(120574 + 120578) (120574 (120583 + 120578) minus 120582119902 (120574 + 120578)) (15)

Under the EPP scheme the server charges the price inproportion to the customerrsquos sojourn time Therefore theexpected benefit has the following relationship119880119905 = 119877minus(119870119905+119862)119882(119902) If 119880119905 equals zero at customerrsquos equilibrium we have


0 0

0 1

1 0

1 1

2 0 n 0

2 1 n 1

middot middot middot



q

q

q

q

q

q

q

q

q

q

Figure 3 Transition diagram for the fully unobservable case

119877 = (119870119905 + 119862)119882(119902) At customerrsquos equilibrium the price canbe expressed in terms of the customerrsquos joining probabilitydenoted by 119902119905119870119905 = 119877119882(119902119905) minus 119862= 119877 (120574 + 120578) (120574 (120583 + 120578) minus 120582119902119905 (120574 + 120578))120574 (120574 + 120578) + 120578 (120583 + 120578) minus 119862 (16)

Substituting (16) into 119875119905 = 120582119902119905119870119905119882(119902119905) we can establish thefollowing NLP problem to maximize 119875119905 with respect to 119902119905

max119902119905

119875119905 = 119877120582119902119905 minus 119862120582119902119905 (120574 (120574 + 120578) + 120578 (120583 + 120578))(120574 + 120578) (120574 (120583 + 120578) minus 120582119902119905 (120574 + 120578))st 0 le 119902119905 le 1120582119902119905 (120578 + 120574) lt 120574 (120583 + 120578)

(17)

In (17) we want to maximize the expected server profit pertime unit under the EPP scheme in the fully unobservablecase The first constraint implies that 119902119905 is the joiningprobability and the second is derived from (11) when 1199020119905 =1199021119905 = 119902119905 We introduce the following

Lemma 8 The optimization problem in (17) is a convexmaximization problem (CMP) In addition the optimal joiningprobability is max0min1199091 1 where 1199091 is the root of theequation 120597119875119905120597119902119905 = 0 satisfying 1205821199091(120578 + 120574) lt 120574(120583 + 120578)Proof See Appendix C

Theorem 9 Under the EPP scheme in the fully unobservablecase if 0 le 119902119905 le 1 there exists a unique equilibrium wherecustomers join the queue with probabilities of 119902lowast119905 = max0min1199091 1 and the optimal price which maximizes theexpected server profit per unit time is given by119870lowast119905 = 119877119882(119902lowast119905 )minus119862Proof The optimal solution 119902lowast119905 of the CMP in (17) is deter-mined by Lemma 8 and the optimal price which maximizesthe expected server profit per unit time is calculated byinserting 119902lowast119905 into (16) This completes the proof

Under the EAP scheme the server charges a flat ratefor services Hence the expected benefit has the following

relationship 119880119891 = 119877 minus 119870119891 minus 119862119882(119902) If 119880119891 equals zeroat customerrsquos equilibrium we have 119877 = 119870119891 + 119862119882(119902) Atcustomerrsquos equilibrium therefore the price can be expressedin terms of the customerrsquos joining probability denoted by 119902119891119870119891 = 119877 minus 119862119882(119902119891)= 119877 minus 119862120574 (120574 + 120578) + 120578 (120583 + 120578)(120574 + 120578) (120574 (120583 + 120578) minus 120582119902119891 (120574 + 120578)) (18)

Inserting (18) into 119875119905 = 120582119902119891119870119891 we have the following NLPproblem to maximize 119875119891 with respect to 119902119891

max119902119891

119875119891= 119877120582119902119891 minus 119862120582119902119891 (120574 (120574 + 120578) + 120578 (120583 + 120578))(120574 + 120578) (120574 (120583 + 120578) minus 120582119902119891 (120574 + 120578))

st 0 le 119902119891 le 1120582119902119891 (120578 + 120574) lt 120574 (120583 + 120578) (19)

Clearly the optimization problem in (19) is identical to thatin (17)

Theorem 10 Under the EAP scheme in the fully unobservablecase if 0 le 119902119891 le 1 there exists a unique equilibrium wherecustomers join the queue with probability 119902lowast119891 = 119902lowast119905 the optimalprice which maximizes the expected server profit per unit timeis given by 119870lowast119891 = 119877 minus 119862119882(119902lowast119891)Proof Note that 119902lowast119891 = 119902lowast119905 from (17) and (19)The optimal pricewhich maximizes the expected server profit per unit time iscalculated by inserting 119902lowast119891 into (18)This completes the proof

Finally we present the numerical analyses The trendsof the optimal price and the joining probability in thefully unobservable case are similar to those in the partiallyunobservable case Instead of conducting the same numericalanalyses we compare the maximum server profits of thetwo cases We set the common experimental parameters as(119877 119862) = (20 2) Let 119875lowastpa and 119875lowastfu denote the maximumserver profit for the partially unobservable case and the fullyunobservable case respectively In Figure 4 we find that 119875lowastpais greater than 119875lowastfu with respect to various values of 120578 and 120582indicating that informing arriving customers of the state ofthe server is more beneficial in terms of the serverrsquos profit

Figure 4(a) tells us that 119875lowastpa increases when 120578 le 120583and decreases when 120578 gt 120583 If the failure rate is greaterthan the service rate more customers will leave the systemdue to server failures resulting in frequent repairs Hencethe sojourn time becomes longer and more customers arereluctant to join the queue This has a negative impact on theserverrsquos profit Moreover 119875lowastfu is not influenced by the value of120578 thus we conclude that 120578 is not a considering factor in thefully unobservable case


PlowastＪ

PlowastＯ

94

95

96

97

98

99

Max

imum

serv

er p

rofit


100

(a) 120582 = 1 120583 = 1 and 120574 = 1

936

938

940

942

Max

imum

serv

er p

rofit


PlowastＪ

PlowastＯ

(b) 120583 = 1 120574 = 1 and 120578 = 005

Figure 4 Maximum server profit for the partially unobservable and the fully unobservable cases

Figure 4(b) shows that120582 has no effect on the serverrsquos profitin both the partially and fully unobservable cases Accordingto our parameter settings 119875lowastpa = 942641 and 119875lowastfu = 935089regardless of the value of 120582This implies that the server alwaysmaintains the same maximum profit by adjusting the joiningprobabilities and the prices for a service even if the value of 120582variesThus 120582 is not a considering factor in both the partiallyand fully unobservable cases

5 Conclusions

In this work we studied customersrsquo joiningbalking behaviorat equilibrium and the serverrsquos optimal pricing strategies inpartially and fully unobservable MM1 queueing systemswith negative customers and repair We introduced two dif-ferent pricing models the EPP scheme and the EAP schemeIn the EPP scheme the server imposes a price that is pro-portional to ordinary customerrsquos sojourn time while in theEAP scheme the server charges a fixed price for the serviceWe set the convex maximization problem and determinedthe optimal joining probabilities and prices that maximizethe serverrsquos expected profits for each pricing scheme We alsodetermined that the two different pricing schemes presentidentical outcomes with regard to customersrsquo equilibriumjoiningbalking behavior and the serverrsquos maximum profitsMoreover through various numerical experiments we foundthat the maximum profit of the server in the partially unob-servable case is greater than that in the fully unobservablecase In conclusion providing information about the state ofthe server to arriving customers is advantageous in terms ofmaximizing the serverrsquos profit

This work could provide queue managers with usefulguidelines for making pricing decisions It can also aidcustomers with regard to optimal strategies Future studiescan include the extension of this work to other queueingsystems with for instance multiple servers a vacation policyandor a sojourn time-dependent reward-cost structure The

MG1-type or GIM1-type pricing analyses can also beconsidered as future research topics

Appendix

A The Proof of Lemma 2

We define the partial probability generating functions119866119894(119911) = suminfin119899=0 119901(119899 119894)119911119899 119894 = 0 1 Note that the normalizingcondition is given by1198660(1)+1198661(1) = 1Multiplying (3)ndash(6) byappropriate powers of 119911 and summing over 119899 ge 0 we obtain

1198660 (119911) = 120578119911minus11198661 (119911) + 120578119901 (0 1) (1 minus 119911minus1)1205821199020 (1 minus 119911) + 120574 1198661 (119911) = 120583119901 (0 1) (1 minus 119911minus1) + 1205741198660 (119911)1205821199021 (1 minus 119911) + 120583 (1 minus 119911minus1) + 120578

(A1)

Solving (A1) with respect to 119866119894(119911) 119894 = 0 1 we have1198660 (119911)= 120578119901 (0 1) (1205821199021 (1 minus 119911) + 120583 + 120578)120574 (120578 + 120583) + 1205821199020 (120583 (1 minus 119911) minus 120578119911) minus 1205821199021119911 (1205821199020 (1 minus 119911) + 120574) 1198661 (119911)= 119901 (0 1) (1205821205831199020 (1 minus 119911) + 120574 (120583 + 120578))120574 (120578 + 120583) + 1205821199020 (120583 (1 minus 119911) minus 120578119911) minus 1205821199021119911 (1205821199020 (1 minus 119911) + 120574) (A2)

By normalizing the only unknown value 119901(0 1) in (A2) isdetermined by119901 (0 1) = 120574 (120583 + 120578) minus 120582 (1205781199020 + 1205741199021)(120574 + 120578) (120583 + 120578) (A3)

Since all states are communicating from the theory ofrecurrent events it can be deduced that the probabilities119901(119899 119894) (119899 ge 0 119894 = 0 1) are either all positive or alternatively


all equal to zero If the Markov chain (119873(119905) 119868(119905)) 119905 ge 0is positive recurrent then all the probabilities 119901(119899 119894) arepositive Therefore 119901(0 1) gt 0 and we have 120582(1205781199020 + 1205741199021) lt120574(120583 + 120578) from (A3) Conversely if 120582(1205781199020 + 1205741199021) lt 120574(120583 + 120578)then 119901(0 1) gt 0 We conclude that all the probabilities 119901(119899 119894)are positive from the ergodicity theory for continuous-timeMarkov chains Thus the vector process (119873(119905) 119868(119905)) 119905 ge 0is stable if and only if 120582(1205781199020 + 1205741199021) lt 120574(120583 + 120578)

Let 1205870 and 1205871 denote the probability that the server isunder repair and in operation respectively By the definitionof 119866119894(119911) we have that 120587119894 = suminfin119899=0 119901(119899 119894) = 119866119894(1) 119894 = 0 1

From Remark 3 1205870 = 1198660(1) = 120578(120574 + 120578)minus1 and 1205871 = 1198661(1) =120574(120574 + 120578)minus1 Therefore we have

119866 (119911) = 1198660 (119911) + 1198661 (119911) = 120574 (120583 + 120578) minus 120582 (1205781199020 + 1205741199021)(120574 + 120578) (120583 + 120578)sdot 120582 (1 minus 119911) (1205781199021 + 1205831199020) + (120574 + 120578) (120583 + 120578)120574 (120578 + 120583) + 1205821199020 (120583 (1 minus 119911) minus 120578119911) minus 1205821199021119911 (1205821199020 (1 minus 119911) + 120574) (A4)Finally using Littlersquos formula the expected sojourn time ofan ordinary customer who decides to join the queue uponhisher arrival is

119882(1199020 1199021) = 1198661015840 (1)120582eff = 1205741199021 (1205821205781199021 + 120574 (120583 + 120578)) + 1205781199020 ((120583 + 120578) (120583 + 120574 + 120578) minus 1205821199021 (120583 + 120574)) + 12058212058312057811990220(120583 + 120578) (1205781199020 + 1205741199021) (120574 (120583 + 120578) minus 120582 (1205781199020 + 1205741199021)) (A5)

B The Proof of Lemma 5

According to the definition of the convex maximization inBoyd and Vandenberghe [36] if the objective function canbe proved concave in the feasible region and the set ofconstraints can be proved convex the maximization problemis a CMP

The constraint functions in (11) are all real-valued linearfunctions therefore the set of constraints is convex

Let H(1199020119905 1199021119905) be the Hessian matrix of the objectivefunction in (11) and it has the form of

H (1199020119905 1199021119905) = ( 120597211987511990512059711990220119905 120597211987511990512059711990201199051205971199021119905120597211987511990512059711990211199051205971199020119905 120597211987511990512059711990221119905 ) (B1)

where120597211987511990512059711990220119905 = minus21198621205822120574120578 ((120583 + 120578 minus 1205821199021119905)2 + 1205821205781199021119905)(120574 (120583 + 120578) minus 120582 (1205781199020119905 + 1205741199021119905))3 120597211987511990512059711990201199051205971199021119905 = 120597211987511990512059711990211199051205971199020119905= minus1198621205822120574120578 (120574 (120583 + 120578 + 1205821199021119905) + 1205821199020119905 (120578 + 2120583 minus 21205821199021119905))(120574 (120583 + 120578) minus 120582 (1205781199020119905 + 1205741199021119905))3

120597211987511990512059711990221119905 = minus21198621205822120574 (1205742 (120583 + 120578) + 1205821205781199020119905 (1205821199020119905 minus 120574))(120574 (120583 + 120578) minus 120582 (1205781199020119905 + 1205741199021119905))3 (B2)

Define 119863119896 as the leading principal minor determinant oforder 119896 forH(1199020119905 1199021119905) Then we have

1198631 = 120597211987511990512059711990220119905 = minus21198621205822120574120578 ((120583 + 120578 minus 1205821199021119905)2 + 1205821205781199021119905)(120574 (120583 + 120578) minus 120582 (1205781199020119905 + 1205741199021119905))3lt 0

1198632 = (120597211987511990512059711990220119905)(120597211987511990512059711990221119905) minus ( 120597211987511990512059711990201199051205971199021119905)2= 119862212058241205742120578 (4120583 + 3120578)(120574 (120583 + 120578) minus 120582 (1205781199020119905 + 1205741199021119905))4 gt 0(B3)

Therefore H(1199020119905 1199021119905) in (B1) is negative definite and theobjective function in (11) should be strictly concave in thefeasible region which leads to (11) being a CMP

C The Proof of Lemma 8

The constraint functions in (17) are all real-valued linearfunctions therefore the set of constraints is convex Thesecond derivative of the objective function in (17) can beexpressed as12059721198751199051205971199022119905 = minus21198621205822 (120583 + 120578) (120574 (120574 + 120578) + 120578 (120583 + 120578))(120574 (120583 + 120578) minus 120582119902119905 (120578 + 120574))3 (C1)

Since 12059721198751199051205971199022119905 lt 0 in the feasible region the objectivefunction 119875119905 should be strictly concave and (17) is a CMP

Let 1199091 and 1199092 be the roots of the equation 120597119875119905120597119902119905 = 0and 1199091 and 1199092 are written as1199091 = 120574 (120583 + 120578)120582 (120578 + 120574)

minus radic1198771198621205741205822 (120583 + 120578) (120578 + 120574)3 (120574 (120574 + 120578) + 120578 (120583 + 120578))1198771205822 (120578 + 120574)3 1199092 = 120574 (120583 + 120578)120582 (120578 + 120574)+ radic1198771198621205741205822 (120583 + 120578) (120578 + 120574)3 (120574 (120574 + 120578) + 120578 (120583 + 120578))1198771205822 (120578 + 120574)3

(C2)

We can verify that 1205821199091(120578 + 120574) lt 120574(120583 + 120578) and 1205821199092(120578 + 120574) gt120574(120583 + 120578) Let 119902lowast119905 be the optimal joining probability Then we


have the following conclusions if 0 le 1199091 le 1 then 119902lowast119905 = 1199091 if1199091 gt 1 then 119902lowast119905 = 1 implying that 119902lowast119905 = max0min1199091 1Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

References

[1] P Naor ldquoThe regulation of queue size by levying tollsrdquo Econo-metrica vol 37 no 1 pp 15ndash24 1969

[2] U Yechiali ldquoOn optimal balking rules and toll charges in theGIM1 queuerdquo Operations Research vol 19 no 2 pp 349ndash3701971

[3] S G Johansen and S Stidham Jr ldquoControl of arrivals to astochastic input-output systemrdquo Advances in Applied Probabil-ity vol 12 no 4 pp 972ndash999 1980

[4] S Stidham Jr ldquoOptimal control of admission to a queueingsystemrdquo Institute of Electrical and Electronics Engineers Trans-actions on Automatic Control vol 30 no 8 pp 705ndash713 1985

[5] H Mendelson and S Whang ldquoOptimal incentive-compatiblepriority pricing for the MM1 queuerdquoOperations Research vol38 no 5 pp 870ndash883 1990

[6] H Chen and M Z Frank ldquoState dependent pricing with aqueuerdquo Institute of Industrial Engineers (IIE) IIE Transactionsvol 33 no 10 pp 847ndash860 2001

[7] C Larsen ldquoInvestigating sensitivity and the impact of infor-mation on pricing decisions in an MM1infin queueing modelrdquoInternational Journal of Production Economics vol 56-57 pp365ndash377 1998

[8] J Erlichman and R Hassin ldquoEquilibrium solutions in theobservable MM1 queue with overtakingrdquo in Proceedings of the4th International ICST Conference on Performance EvaluationMethodologies and Tools (VALUETOOLS rsquo09) article no 64ICST (Institute for Computer Sciences Social-Informatics andTelecommunications Engineering) Pisa Italy October 2009

[9] N M Edelson and D K Hildebrand ldquoCongestion tolls forPoisson queuing processesrdquo Econometrica vol 43 no 1 pp 81ndash92 1975

[10] R Hassin and M Haviv ldquoEquilibrium threshold strategies Thecase of queues with prioritiesrdquo Operations Research vol 45 no6 pp 966ndash973 1997

[11] R Hassin andM Haviv To Queue or Not to Queue EquilibriumBehavior in Queueing Systems International Series in Oper-ations Research amp Management Science Kluwer AcademicPublishers Dordrecht the Netherlands 2003

[12] E Gelenbe ldquoRandom neural networks with negative and pos-itive signals and product form solutionrdquo Neural Computationvol 1 no 4 pp 502ndash510 1989

[13] E Gelenbe ldquoG-networks a unifying model for neural andqueueing networksrdquo Annals of Operations Research vol 48 no5 pp 433ndash461 1994

[14] J R Artalejo ldquoG-networks A versatile approach for workremoval in queueing networksrdquo European Journal of Opera-tional Research vol 126 no 2 pp 233ndash249 2000

[15] K C Chae H M Park and W S Yang ldquoA GIGeo1 queuewith negative and positive customersrdquo Applied MathematicalModelling vol 34 no 6 pp 1662ndash1671 2010

[16] D H Lee W S Yang and H M Park ldquoGeoG1 queueswith disasters and general repair timesrdquo Applied MathematicalModelling vol 35 no 4 pp 1561ndash1570 2011

[17] A Economou and S Kanta ldquoEquilibrium balking strategiesin the observable single-server queue with breakdowns andrepairsrdquoOperations Research Letters vol 36 no 6 pp 696ndash6992008

[18] O Boudali and A Economou ldquoOptimal and equilibriumbalking strategies in the single server Markovian queue withcatastrophesrdquo European Journal of Operational Research vol218 no 3 pp 708ndash715 2012

[19] O Boudali and A Economou ldquoThe effect of catastrophes onthe strategic customer behavior in queueing systemsrdquo NavalResearch Logistics (NRL) vol 60 no 7 pp 571ndash587 2013

[20] A Economou and A Manou ldquoEquilibrium balking strategiesfor a clearing queueing system in alternating environmentrdquoAnnals of Operations Research vol 208 no 1 pp 489ndash514 2013

[21] F Zhang J Wang and B Liu ldquoOn the optimal and equilibriumretrial rates in an unreliable retrial queue with vacationsrdquoJournal of Industrial and Management Optimization vol 8 no4 pp 861ndash875 2012

[22] F Wang J Wang and F Zhang ldquoStrategic behavior in thesingle-server constant retrial queue with individual removalrdquoQuality Technology and Quantitative Management vol 12 no3 pp 325ndash342 2015

[23] L Li J Wang and F Zhang ldquoEquilibrium customer strategiesin Markovian queues with partial breakdownsrdquo Computers ampIndustrial Engineering vol 66 no 4 pp 751ndash757 2013

[24] J Wang and F Zhang ldquoEquilibrium analysis of the observablequeues with balking and delayed repairsrdquo Applied Mathematicsand Computation vol 218 no 6 pp 2716ndash2729 2011

[25] S Yu Z Liu and JWu ldquoEquilibrium strategies of the unobserv-able MM1 queue with balking and delayed repairsrdquo AppliedMathematics and Computation vol 290 pp 56ndash65 2016

[26] A Veltman and R Hassin ldquoEquilibrium in queueing systemswith complementary productsrdquo Queueing Systems vol 50 no2-3 pp 325ndash342 2005

[27] W Sun S-y Li N-s Tian and H-k Zhang ldquoEquilibriumanalysis in batch-arrival queues with complementary servicesrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 33 no 1 pp224ndash241 2009

[28] A Economou and S Kanta ldquoEquilibrium customer strategiesand social-profit maximization in the single-server constantretrial queuerdquoNaval Research Logistics (NRL) vol 58 no 2 pp107ndash122 2011

[29] J Wang and F Zhang ldquoMonopoly pricing in a retrial queuewith delayed vacations for local area network applicationsrdquo IMAJournal of Management Mathematics vol 27 no 2 pp 315ndash3342016

[30] Y Zhang J Wang and FWang ldquoEquilibrium pricing strategiesin retrial queueing systems with complementary servicesrdquoApplied Mathematical Modelling Simulation and Computationfor Engineering and Environmental Systems vol 40 no 11-12 pp5775ndash5792 2016

[31] YMa andZ Liu ldquoPricing analysis inGEOGEO1 queueing sys-temrdquo Mathematical Problems in Engineering vol 2015 ArticleID 181653 5 pages 2015

[32] D H Lee ldquoA note on the optimal pricing strategy in thediscrete-time GeoGeo1 queuing system with sojourn time-dependent rewardrdquo Operations Research Perspectives vol 4 pp113ndash117 2017


[33] B Yang F Tan and Y S Dai ldquoPerformance evaluation of cloudservice considering fault recoveryrdquoThe Journal of Supercomput-ing vol 65 no 1 pp 426ndash444 2013

[34] J Cao KHwang K Li andA Y Zomaya ldquoOptimalmultiserverconfiguration for profit maximization in cloud computingrdquoIEEE Transactions on Parallel and Distributed Systems vol 24no 6 pp 1087ndash1096 2013

[35] R W Wolff ldquoPoisson arrivals see time averagesrdquo OperationsResearch The Journal of the Operations Research Society ofAmerica vol 30 no 2 pp 223ndash231 1982

[36] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press 2004

Submit your manuscripts athttpswwwhindawicom

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Stochastic AnalysisInternational Journal of


servicemanufacturing systems If a negative customer arrivesat a system it removes one ordinary customer (called apositive customer) according to a predetermined removaldiscipline and the server breaks down Because the perfor-mance measure of such a systemmay be influenced by serverfailure and repair operations such an unreliable system iswell worth analyzing in light of both queueing theory andreliability theory Additional studies on unreliable queueingsystems were conducted by Gelenbe [13] Artalejo [14] Chaeet al [15] and Lee et al [16] and references therein

However few works have attempted an equilibriumanalysis of an unreliable queueing system despite the factthat server failure is a common phenomenon in everydaylife For example equilibrium strategies with various levelsof information were studied by Economou and Kanta [17]Boudali and Economou [18 19] and Economou and Manou[20] considered equilibrium balking strategies in an unre-liable single-server queue with catastrophes (or stochasticclearing) where the server fails and all present customersin the system are forced to leave upon the catastrophersquosarrival Zhang et al [21] analyzed optimal and equilibriumretrial rates in an unreliable queue in which the server issubject to vacations along with breakdowns and repair Wanget al [22] studied the equilibrium behavior of customersin a retrial queue with negative customers and derivedcorresponding Nash equilibrium joining strategies and asocial net benefit maximization problem with respect to thelevels of information available to customers upon arrival Liet al [23] conducted an equilibrium analysis on unreliableMM1 queueing systems with working breakdowns Whenthe system is under repair the secondary (substitute) serverprovides services to customers at a slow speed instead of stop-ping servicing completelyWang andZhang [24] extended themodel in [17] by assuming that the failed server experiencesan exponential delay before the repair begins Yu et al [25]also dealt with the same model of [24] for an unobservablecase

Veltman and Hassin [26] presented an interesting paperon the pricing strategies of a server in a Markovian MM1queueing system They analyzed equilibrium pricing strate-gies under two types of pricing structures In the first pricingstructure the server charges a fee that is proportional tothe sojourn time (the waiting time plus the service time)referred to as an ex-post payment (EPP) scheme In thesecond however the server charges a fixed fee for theservice meaning that the server implements an ex-ante pay-ment (EAP) scheme Under each pricing scheme the equi-librium joining behavior of customers was examined andcompared with socially optimal results Later Sun et al [27]extended this analysis to a batch arrival queueing system andEconomou and Kanta [28]Wang and Zhang [29] and Zhanget al [30] studied retrial cases Recently Ma and Liu [31]considered pricing strategies in a discrete-time GeoGeo1queueing system and Lee [32] studied the optimal pricingstrategies in a discrete-timeGeoGeo1 queuing systemunderthe EPP and EAP schemes by assuming the sojourn time-dependent reward

A queueing system with negative customers can be foundin many real-life systems One example is a cloud data center

where the scheduler allocates a virtual machine (VM) fora service request which is generated by a user All arrivingservice requests are initially buffered in the scheduler queueafter which each is transmitted to a certain VM according toa predetermined scheduling policy built into the schedulerIn addition the scheduler oversees service request processingand the VM status for the billing of services Dependingon the operating conditions of the cloud data center thescheduler charges a fixed price regardless of the system usagetime or charges a price that is proportional to the timeTherefore the scheduler plays a key role in operating thecloud data center stably andmaximizing revenue Because thescheduler is the single point of failure of the entire cloud datacenter a secondary scheduler must be built in preparation fora system failure network hacking or high availability Oncethemain scheduler fails the request in service is deleted fromthe scheduler and the secondary scheduler replaces the mainscheduler While the main scheduler is being repaired thesecondary scheduler stores the arriving service requests onlyin the scheduler queueTherefore in terms ofmaximizing theoperation profits of the cloud data center analyzing optimaland equilibrium pricing strategies is important

Motivated by the aforementioned studies and the possibleapplication to pricing in a cloud data center we analyze opti-mal pricing strategies and the equilibrium joiningbalkingbehavior of customers in an unobservable MM1 queue-ing system with negative customers and repair under thetwo pricing schemes described above Our contributionsare twofold First we derive not only the optimal joiningstrategies of customers but also the optimal prices set bythe server in equilibrium under our model To the authorrsquosknowledge this study is the first to consider the pricingstrategies together with the customer equilibrium behaviorfor an MM1 queueing system with server failure and repairSecond under the EPP and EAP schemes we show thatthe two pricing schemes do not affect customersrsquo joiningbehavior and the serverrsquos maximum profits which has notbeen reported in the literature on anMM1 queueing systemwith negative customers and repair

This paper is structured as follows In Section 2 wedescribe the model and derive the expected sojourn timeof an unobservable MM1 queueing system with negativecustomers and repair In Sections 3 and 4 customersrsquo equi-librium joining behavior and the serverrsquos profit maximizationstrategies under the two pricing schemes are driven in a par-tially unobservable case and in a fully unobservable case Wealso deal with numerical experiments in which we investigatetrends in the optimal prices and joining probabilities of EPPand EAP schemes according to various input values

2 Preliminaries

This work considers a queueing system with the followingfeatures Ordinary customers arrive at a single server queueaccording to a Poisson process with a rate of 120582 (120582 gt 0)A server renders a service to each ordinary customer on afirst-come first-out basis The service time is exponentiallydistributed with a rate of 120583 (120583 gt 0) Negative customersarrive only when the server is in operation according to a








(2)



(1205821199021 + 120578) 119901 (0 1) = 120574119901 (0 0) + 120583119901 (1 1) (5)(1205821199021 + 120583 + 120578) 119901 (119899 1) = 120574119901 (119899 0) + 1205821199021119901 (119899 minus 1 1)+ 120583119901 (119899 + 1 1) 119899 ge 1 (6)



119882(1199020 1199021) = 1205741199021 (1205821205781199021 + 120574 (120583 + 120578)) + 1205781199020 ((120583 + 120578) (120583 + 120574 + 120578) minus 1205821199021 (120583 + 120574)) + 12058212058312057811990220(120583 + 120578) (1205781199020 + 1205741199021) (120574 (120583 + 120578) minus 120582 (1205781199020 + 1205741199021)) (7)






0 0

0 1

1 0

1 1

2 0 n 0

2 1 n 1

q1q1q1q1q1

q0q0q0q0q0








Q = (minus120574 120574120578 minus120578) (8)





max1199020119905 1199021119905

119875119905 = 120582 (1199020119905120578 + 1199021119905120574) (119877 minus 119862119882(1199020119905 1199021119905))120574 + 120578st 0 le 119902119894119905 le 1 119894 = 0 1120582 (1205781199020119905 + 1205741199021119905) lt 120574 (120583 + 120578)

(11)














= 120582 (1199020119891120578 + 1199021119891120574) (119877 minus 119862119882(1199020119891 1199021119891))120574 + 120578 (13)


max1199020119891 1199021119891

119875119891= 120582 (1199020119891120578 + 1199021119891120574) (119877 minus 119862119882(1199020119891 1199021119891))120574 + 120578

st 0 le 119902119894119891 le 1 119894 = 0 1120582 (1205781199020119891 + 1205741199021119891) lt 120574 (120583 + 120578) (14)






04

05

06

07

08

Opt

imal

join

ing

prob

abili

ty

09 10 11 1208Service rate ()

qlowast1

qlowast0

(a) 120582 = 1 120574 = 15 and 120578 = 005

02

04

06

08

10

Opt

imal

join

ing

prob

abili

ty

10 15 20 25 30 35 4005Repair rate ()

qlowast1

qlowast0

(b) 120582 = 1 120583 = 1 and 120578 = 05

02

04

06

08

10

Opt

imal

join

ing

prob

abili

ty


qlowast1

qlowast0

(c) 120582 = 1 120583 = 1 and 120574 = 1

02

04

06

08

Opt

imal

join

ing

prob

abili

ty


qlowast1

qlowast0

(d) 120583 = 1 120574 = 1 and 120578 = 005








119882(119902) = 120574 (120574 + 120578) + 120578 (120583 + 120578)(120574 + 120578) (120574 (120583 + 120578) minus 120582119902 (120574 + 120578)) (15)



0 0

0 1

1 0

1 1

2 0 n 0

2 1 n 1




q

q

q

q

q

q

q

q

q

q




max119902119905


(17)







max119902119891

119875119891= 119877120582119902119891 minus 119862120582119902119891 (120574 (120574 + 120578) + 120578 (120583 + 120578))(120574 + 120578) (120574 (120583 + 120578) minus 120582119902119891 (120574 + 120578))

st 0 le 119902119891 le 1120582119902119891 (120578 + 120574) lt 120574 (120583 + 120578) (19)






PlowastＪ

PlowastＯ

94

95

96

97

98

99

Max

imum

serv

er p

rofit


100

(a) 120582 = 1 120583 = 1 and 120574 = 1

936

938

940

942

Max

imum

serv

er p

rofit


PlowastＪ

PlowastＯ

(b) 120583 = 1 120574 = 1 and 120578 = 005



5 Conclusions




Appendix




(A1)









119882(1199020 1199021) = 1198661015840 (1)120582eff = 1205741199021 (1205821205781199021 + 120574 (120583 + 120578)) + 1205781199020 ((120583 + 120578) (120583 + 120574 + 120578) minus 1205821199021 (120583 + 120574)) + 12058212058312057811990220(120583 + 120578) (1205781199020 + 1205741199021) (120574 (120583 + 120578) minus 120582 (1205781199020 + 1205741199021)) (A5)





H (1199020119905 1199021119905) = ( 120597211987511990512059711990220119905 120597211987511990512059711990201199051205971199021119905120597211987511990512059711990211199051205971199020119905 120597211987511990512059711990221119905 ) (B1)





1198632 = (120597211987511990512059711990220119905)(120597211987511990512059711990221119905) minus ( 120597211987511990512059711990201199051205971199021119905)2= 119862212058241205742120578 (4120583 + 3120578)(120574 (120583 + 120578) minus 120582 (1205781199020119905 + 1205741199021119905))4 gt 0(B3)







(C2)





References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











(2)



(1205821199021 + 120578) 119901 (0 1) = 120574119901 (0 0) + 120583119901 (1 1) (5)(1205821199021 + 120583 + 120578) 119901 (119899 1) = 120574119901 (119899 0) + 1205821199021119901 (119899 minus 1 1)+ 120583119901 (119899 + 1 1) 119899 ge 1 (6)



119882(1199020 1199021) = 1205741199021 (1205821205781199021 + 120574 (120583 + 120578)) + 1205781199020 ((120583 + 120578) (120583 + 120574 + 120578) minus 1205821199021 (120583 + 120574)) + 12058212058312057811990220(120583 + 120578) (1205781199020 + 1205741199021) (120574 (120583 + 120578) minus 120582 (1205781199020 + 1205741199021)) (7)






0 0

0 1

1 0

1 1

2 0 n 0

2 1 n 1

q1q1q1q1q1

q0q0q0q0q0








Q = (minus120574 120574120578 minus120578) (8)





max1199020119905 1199021119905

119875119905 = 120582 (1199020119905120578 + 1199021119905120574) (119877 minus 119862119882(1199020119905 1199021119905))120574 + 120578st 0 le 119902119894119905 le 1 119894 = 0 1120582 (1205781199020119905 + 1205741199021119905) lt 120574 (120583 + 120578)

(11)














= 120582 (1199020119891120578 + 1199021119891120574) (119877 minus 119862119882(1199020119891 1199021119891))120574 + 120578 (13)


max1199020119891 1199021119891

119875119891= 120582 (1199020119891120578 + 1199021119891120574) (119877 minus 119862119882(1199020119891 1199021119891))120574 + 120578

st 0 le 119902119894119891 le 1 119894 = 0 1120582 (1205781199020119891 + 1205741199021119891) lt 120574 (120583 + 120578) (14)






04

05

06

07

08

Opt

imal

join

ing

prob

abili

ty

09 10 11 1208Service rate ()

qlowast1

qlowast0

(a) 120582 = 1 120574 = 15 and 120578 = 005

02

04

06

08

10

Opt

imal

join

ing

prob

abili

ty

10 15 20 25 30 35 4005Repair rate ()

qlowast1

qlowast0

(b) 120582 = 1 120583 = 1 and 120578 = 05

02

04

06

08

10

Opt

imal

join

ing

prob

abili

ty


qlowast1

qlowast0

(c) 120582 = 1 120583 = 1 and 120574 = 1

02

04

06

08

Opt

imal

join

ing

prob

abili

ty


qlowast1

qlowast0

(d) 120583 = 1 120574 = 1 and 120578 = 005








119882(119902) = 120574 (120574 + 120578) + 120578 (120583 + 120578)(120574 + 120578) (120574 (120583 + 120578) minus 120582119902 (120574 + 120578)) (15)



0 0

0 1

1 0

1 1

2 0 n 0

2 1 n 1




q

q

q

q

q

q

q

q

q

q




max119902119905


(17)







max119902119891

119875119891= 119877120582119902119891 minus 119862120582119902119891 (120574 (120574 + 120578) + 120578 (120583 + 120578))(120574 + 120578) (120574 (120583 + 120578) minus 120582119902119891 (120574 + 120578))

st 0 le 119902119891 le 1120582119902119891 (120578 + 120574) lt 120574 (120583 + 120578) (19)






PlowastＪ

PlowastＯ

94

95

96

97

98

99

Max

imum

serv

er p

rofit


100

(a) 120582 = 1 120583 = 1 and 120574 = 1

936

938

940

942

Max

imum

serv

er p

rofit


PlowastＪ

PlowastＯ

(b) 120583 = 1 120574 = 1 and 120578 = 005



5 Conclusions




Appendix




(A1)









119882(1199020 1199021) = 1198661015840 (1)120582eff = 1205741199021 (1205821205781199021 + 120574 (120583 + 120578)) + 1205781199020 ((120583 + 120578) (120583 + 120574 + 120578) minus 1205821199021 (120583 + 120574)) + 12058212058312057811990220(120583 + 120578) (1205781199020 + 1205741199021) (120574 (120583 + 120578) minus 120582 (1205781199020 + 1205741199021)) (A5)





H (1199020119905 1199021119905) = ( 120597211987511990512059711990220119905 120597211987511990512059711990201199051205971199021119905120597211987511990512059711990211199051205971199020119905 120597211987511990512059711990221119905 ) (B1)





1198632 = (120597211987511990512059711990220119905)(120597211987511990512059711990221119905) minus ( 120597211987511990512059711990201199051205971199021119905)2= 119862212058241205742120578 (4120583 + 3120578)(120574 (120583 + 120578) minus 120582 (1205781199020119905 + 1205741199021119905))4 gt 0(B3)







(C2)





References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





0 0

0 1

1 0

1 1

2 0 n 0

2 1 n 1

q1q1q1q1q1

q0q0q0q0q0








Q = (minus120574 120574120578 minus120578) (8)





max1199020119905 1199021119905

119875119905 = 120582 (1199020119905120578 + 1199021119905120574) (119877 minus 119862119882(1199020119905 1199021119905))120574 + 120578st 0 le 119902119894119905 le 1 119894 = 0 1120582 (1205781199020119905 + 1205741199021119905) lt 120574 (120583 + 120578)

(11)














= 120582 (1199020119891120578 + 1199021119891120574) (119877 minus 119862119882(1199020119891 1199021119891))120574 + 120578 (13)


max1199020119891 1199021119891

119875119891= 120582 (1199020119891120578 + 1199021119891120574) (119877 minus 119862119882(1199020119891 1199021119891))120574 + 120578

st 0 le 119902119894119891 le 1 119894 = 0 1120582 (1205781199020119891 + 1205741199021119891) lt 120574 (120583 + 120578) (14)






04

05

06

07

08

Opt

imal

join

ing

prob

abili

ty

09 10 11 1208Service rate ()

qlowast1

qlowast0

(a) 120582 = 1 120574 = 15 and 120578 = 005

02

04

06

08

10

Opt

imal

join

ing

prob

abili

ty

10 15 20 25 30 35 4005Repair rate ()

qlowast1

qlowast0

(b) 120582 = 1 120583 = 1 and 120578 = 05

02

04

06

08

10

Opt

imal

join

ing

prob

abili

ty


qlowast1

qlowast0

(c) 120582 = 1 120583 = 1 and 120574 = 1

02

04

06

08

Opt

imal

join

ing

prob

abili

ty


qlowast1

qlowast0

(d) 120583 = 1 120574 = 1 and 120578 = 005








119882(119902) = 120574 (120574 + 120578) + 120578 (120583 + 120578)(120574 + 120578) (120574 (120583 + 120578) minus 120582119902 (120574 + 120578)) (15)



0 0

0 1

1 0

1 1

2 0 n 0

2 1 n 1




q

q

q

q

q

q

q

q

q

q




max119902119905


(17)







max119902119891

119875119891= 119877120582119902119891 minus 119862120582119902119891 (120574 (120574 + 120578) + 120578 (120583 + 120578))(120574 + 120578) (120574 (120583 + 120578) minus 120582119902119891 (120574 + 120578))

st 0 le 119902119891 le 1120582119902119891 (120578 + 120574) lt 120574 (120583 + 120578) (19)






PlowastＪ

PlowastＯ

94

95

96

97

98

99

Max

imum

serv

er p

rofit


100

(a) 120582 = 1 120583 = 1 and 120574 = 1

936

938

940

942

Max

imum

serv

er p

rofit


PlowastＪ

PlowastＯ

(b) 120583 = 1 120574 = 1 and 120578 = 005



5 Conclusions




Appendix




(A1)









119882(1199020 1199021) = 1198661015840 (1)120582eff = 1205741199021 (1205821205781199021 + 120574 (120583 + 120578)) + 1205781199020 ((120583 + 120578) (120583 + 120574 + 120578) minus 1205821199021 (120583 + 120574)) + 12058212058312057811990220(120583 + 120578) (1205781199020 + 1205741199021) (120574 (120583 + 120578) minus 120582 (1205781199020 + 1205741199021)) (A5)





H (1199020119905 1199021119905) = ( 120597211987511990512059711990220119905 120597211987511990512059711990201199051205971199021119905120597211987511990512059711990211199051205971199020119905 120597211987511990512059711990221119905 ) (B1)





1198632 = (120597211987511990512059711990220119905)(120597211987511990512059711990221119905) minus ( 120597211987511990512059711990201199051205971199021119905)2= 119862212058241205742120578 (4120583 + 3120578)(120574 (120583 + 120578) minus 120582 (1205781199020119905 + 1205741199021119905))4 gt 0(B3)







(C2)





References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of












= 120582 (1199020119891120578 + 1199021119891120574) (119877 minus 119862119882(1199020119891 1199021119891))120574 + 120578 (13)


max1199020119891 1199021119891

119875119891= 120582 (1199020119891120578 + 1199021119891120574) (119877 minus 119862119882(1199020119891 1199021119891))120574 + 120578

st 0 le 119902119894119891 le 1 119894 = 0 1120582 (1205781199020119891 + 1205741199021119891) lt 120574 (120583 + 120578) (14)






04

05

06

07

08

Opt

imal

join

ing

prob

abili

ty

09 10 11 1208Service rate ()

qlowast1

qlowast0

(a) 120582 = 1 120574 = 15 and 120578 = 005

02

04

06

08

10

Opt

imal

join

ing

prob

abili

ty

10 15 20 25 30 35 4005Repair rate ()

qlowast1

qlowast0

(b) 120582 = 1 120583 = 1 and 120578 = 05

02

04

06

08

10

Opt

imal

join

ing

prob

abili

ty


qlowast1

qlowast0

(c) 120582 = 1 120583 = 1 and 120574 = 1

02

04

06

08

Opt

imal

join

ing

prob

abili

ty


qlowast1

qlowast0

(d) 120583 = 1 120574 = 1 and 120578 = 005








119882(119902) = 120574 (120574 + 120578) + 120578 (120583 + 120578)(120574 + 120578) (120574 (120583 + 120578) minus 120582119902 (120574 + 120578)) (15)



0 0

0 1

1 0

1 1

2 0 n 0

2 1 n 1




q

q

q

q

q

q

q

q

q

q




max119902119905


(17)







max119902119891

119875119891= 119877120582119902119891 minus 119862120582119902119891 (120574 (120574 + 120578) + 120578 (120583 + 120578))(120574 + 120578) (120574 (120583 + 120578) minus 120582119902119891 (120574 + 120578))

st 0 le 119902119891 le 1120582119902119891 (120578 + 120574) lt 120574 (120583 + 120578) (19)






PlowastＪ

PlowastＯ

94

95

96

97

98

99

Max

imum

serv

er p

rofit


100

(a) 120582 = 1 120583 = 1 and 120574 = 1

936

938

940

942

Max

imum

serv

er p

rofit


PlowastＪ

PlowastＯ

(b) 120583 = 1 120574 = 1 and 120578 = 005



5 Conclusions




Appendix




(A1)









119882(1199020 1199021) = 1198661015840 (1)120582eff = 1205741199021 (1205821205781199021 + 120574 (120583 + 120578)) + 1205781199020 ((120583 + 120578) (120583 + 120574 + 120578) minus 1205821199021 (120583 + 120574)) + 12058212058312057811990220(120583 + 120578) (1205781199020 + 1205741199021) (120574 (120583 + 120578) minus 120582 (1205781199020 + 1205741199021)) (A5)





H (1199020119905 1199021119905) = ( 120597211987511990512059711990220119905 120597211987511990512059711990201199051205971199021119905120597211987511990512059711990211199051205971199020119905 120597211987511990512059711990221119905 ) (B1)





1198632 = (120597211987511990512059711990220119905)(120597211987511990512059711990221119905) minus ( 120597211987511990512059711990201199051205971199021119905)2= 119862212058241205742120578 (4120583 + 3120578)(120574 (120583 + 120578) minus 120582 (1205781199020119905 + 1205741199021119905))4 gt 0(B3)







(C2)





References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





04

05

06

07

08

Opt

imal

join

ing

prob

abili

ty

09 10 11 1208Service rate ()

qlowast1

qlowast0

(a) 120582 = 1 120574 = 15 and 120578 = 005

02

04

06

08

10

Opt

imal

join

ing

prob

abili

ty

10 15 20 25 30 35 4005Repair rate ()

qlowast1

qlowast0

(b) 120582 = 1 120583 = 1 and 120578 = 05

02

04

06

08

10

Opt

imal

join

ing

prob

abili

ty


qlowast1

qlowast0

(c) 120582 = 1 120583 = 1 and 120574 = 1

02

04

06

08

Opt

imal

join

ing

prob

abili

ty


qlowast1

qlowast0

(d) 120583 = 1 120574 = 1 and 120578 = 005








119882(119902) = 120574 (120574 + 120578) + 120578 (120583 + 120578)(120574 + 120578) (120574 (120583 + 120578) minus 120582119902 (120574 + 120578)) (15)



0 0

0 1

1 0

1 1

2 0 n 0

2 1 n 1




q

q

q

q

q

q

q

q

q

q




max119902119905


(17)







max119902119891

119875119891= 119877120582119902119891 minus 119862120582119902119891 (120574 (120574 + 120578) + 120578 (120583 + 120578))(120574 + 120578) (120574 (120583 + 120578) minus 120582119902119891 (120574 + 120578))

st 0 le 119902119891 le 1120582119902119891 (120578 + 120574) lt 120574 (120583 + 120578) (19)






PlowastＪ

PlowastＯ

94

95

96

97

98

99

Max

imum

serv

er p

rofit


100

(a) 120582 = 1 120583 = 1 and 120574 = 1

936

938

940

942

Max

imum

serv

er p

rofit


PlowastＪ

PlowastＯ

(b) 120583 = 1 120574 = 1 and 120578 = 005



5 Conclusions




Appendix




(A1)









119882(1199020 1199021) = 1198661015840 (1)120582eff = 1205741199021 (1205821205781199021 + 120574 (120583 + 120578)) + 1205781199020 ((120583 + 120578) (120583 + 120574 + 120578) minus 1205821199021 (120583 + 120574)) + 12058212058312057811990220(120583 + 120578) (1205781199020 + 1205741199021) (120574 (120583 + 120578) minus 120582 (1205781199020 + 1205741199021)) (A5)





H (1199020119905 1199021119905) = ( 120597211987511990512059711990220119905 120597211987511990512059711990201199051205971199021119905120597211987511990512059711990211199051205971199020119905 120597211987511990512059711990221119905 ) (B1)





1198632 = (120597211987511990512059711990220119905)(120597211987511990512059711990221119905) minus ( 120597211987511990512059711990201199051205971199021119905)2= 119862212058241205742120578 (4120583 + 3120578)(120574 (120583 + 120578) minus 120582 (1205781199020119905 + 1205741199021119905))4 gt 0(B3)







(C2)





References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





0 0

0 1

1 0

1 1

2 0 n 0

2 1 n 1




q

q

q

q

q

q

q

q

q

q




max119902119905


(17)







max119902119891

119875119891= 119877120582119902119891 minus 119862120582119902119891 (120574 (120574 + 120578) + 120578 (120583 + 120578))(120574 + 120578) (120574 (120583 + 120578) minus 120582119902119891 (120574 + 120578))

st 0 le 119902119891 le 1120582119902119891 (120578 + 120574) lt 120574 (120583 + 120578) (19)






PlowastＪ

PlowastＯ

94

95

96

97

98

99

Max

imum

serv

er p

rofit


100

(a) 120582 = 1 120583 = 1 and 120574 = 1

936

938

940

942

Max

imum

serv

er p

rofit


PlowastＪ

PlowastＯ

(b) 120583 = 1 120574 = 1 and 120578 = 005



5 Conclusions




Appendix




(A1)









119882(1199020 1199021) = 1198661015840 (1)120582eff = 1205741199021 (1205821205781199021 + 120574 (120583 + 120578)) + 1205781199020 ((120583 + 120578) (120583 + 120574 + 120578) minus 1205821199021 (120583 + 120574)) + 12058212058312057811990220(120583 + 120578) (1205781199020 + 1205741199021) (120574 (120583 + 120578) minus 120582 (1205781199020 + 1205741199021)) (A5)





H (1199020119905 1199021119905) = ( 120597211987511990512059711990220119905 120597211987511990512059711990201199051205971199021119905120597211987511990512059711990211199051205971199020119905 120597211987511990512059711990221119905 ) (B1)





1198632 = (120597211987511990512059711990220119905)(120597211987511990512059711990221119905) minus ( 120597211987511990512059711990201199051205971199021119905)2= 119862212058241205742120578 (4120583 + 3120578)(120574 (120583 + 120578) minus 120582 (1205781199020119905 + 1205741199021119905))4 gt 0(B3)







(C2)





References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





PlowastＪ

PlowastＯ

94

95

96

97

98

99

Max

imum

serv

er p

rofit


100

(a) 120582 = 1 120583 = 1 and 120574 = 1

936

938

940

942

Max

imum

serv

er p

rofit


PlowastＪ

PlowastＯ

(b) 120583 = 1 120574 = 1 and 120578 = 005



5 Conclusions




Appendix




(A1)









119882(1199020 1199021) = 1198661015840 (1)120582eff = 1205741199021 (1205821205781199021 + 120574 (120583 + 120578)) + 1205781199020 ((120583 + 120578) (120583 + 120574 + 120578) minus 1205821199021 (120583 + 120574)) + 12058212058312057811990220(120583 + 120578) (1205781199020 + 1205741199021) (120574 (120583 + 120578) minus 120582 (1205781199020 + 1205741199021)) (A5)





H (1199020119905 1199021119905) = ( 120597211987511990512059711990220119905 120597211987511990512059711990201199051205971199021119905120597211987511990512059711990211199051205971199020119905 120597211987511990512059711990221119905 ) (B1)





1198632 = (120597211987511990512059711990220119905)(120597211987511990512059711990221119905) minus ( 120597211987511990512059711990201199051205971199021119905)2= 119862212058241205742120578 (4120583 + 3120578)(120574 (120583 + 120578) minus 120582 (1205781199020119905 + 1205741199021119905))4 gt 0(B3)







(C2)





References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of









119882(1199020 1199021) = 1198661015840 (1)120582eff = 1205741199021 (1205821205781199021 + 120574 (120583 + 120578)) + 1205781199020 ((120583 + 120578) (120583 + 120574 + 120578) minus 1205821199021 (120583 + 120574)) + 12058212058312057811990220(120583 + 120578) (1205781199020 + 1205741199021) (120574 (120583 + 120578) minus 120582 (1205781199020 + 1205741199021)) (A5)





H (1199020119905 1199021119905) = ( 120597211987511990512059711990220119905 120597211987511990512059711990201199051205971199021119905120597211987511990512059711990211199051205971199020119905 120597211987511990512059711990221119905 ) (B1)





1198632 = (120597211987511990512059711990220119905)(120597211987511990512059711990221119905) minus ( 120597211987511990512059711990201199051205971199021119905)2= 119862212058241205742120578 (4120583 + 3120578)(120574 (120583 + 120578) minus 120582 (1205781199020119905 + 1205741199021119905))4 gt 0(B3)







(C2)





References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




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