research article markovian queueing system with...

Research ArticleMarkovian Queueing System withDiscouraged Arrivals and Self-Regulatory Servers

K. V. Abdul Rasheed and M. Manoharan

Department of Statistics, University of Calicut, Kerala 673 635, India

Correspondence should be addressed to K. V. Abdul Rasheed; [email protected]

Received 28 September 2015; Revised 1 February 2016; Accepted 28 March 2016

Academic Editor: Ahmed Ghoniem

Copyright © 2016 K. V. Abdul Rasheed and M. Manoharan.This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

We consider discouraged arrival of Markovian queueing systems whose service speed is regulated according to the number ofcustomers in the system. We will reduce the congestion in two ways. First we attempt to reduce the congestion by discouraging thearrivals of customers from joining the queue. Secondly we reduce the congestion by introducing the concept of service switches.First we consider a model in which multiple servers have three service rates 𝜇

1, 𝜇2, and 𝜇 (𝜇

1≤ 𝜇2

< 𝜇), say, slow, medium,and fast rates, respectively. If the number of customers in the system exceeds a particular point 𝐾

1or 𝐾2, the server switches to the

medium or fast rate, respectively. For this adaptive queueing system the steady state probabilities are derived and some performancemeasures such as expected number in the system/queue and expected waiting time in the system/queue are obtained. Multipleserver discouraged arrival model having one service switch and single server discouraged arrival model having one and two serviceswitches are obtained as special cases. A Matlab program of the model is presented and numerical illustrations are given.

1. Introduction

Queueing theory plays an important role in modeling reallife problems involving congestion in wide areas of appliedsciences. A customer decides to join the queue only whena short wait is expected and if the wait has been sufficientlysmall he tends to remain in the queue; otherwise the customerleaves the system and then the customer is said to beimpatient. When this impatience increases and customersleave before being served, some remedial actions must betaken to reduce the congestion in the system.

Balking and reneging are the forms of impatience. If acustomer decides not to enter the queue upon arrival byseeing a long queue, the customer is said to have balked. Acustomer may enter the queue but after a while loses patienceand decides to leave and then the customer is said to bereneged.

In the study of queueing system the server is usuallyassumed to work at constant speed regardless of the amountof work existing. But in real life situation this assumptionmaynot always be appropriate as the system size may affect thesystem performance. That is, the servers adapt to the system

state by increasing the speed to clear the queue or decrease thespeed when fatigued, which means the service rate dependson the system size; see, for example, Jonckheere and Borst [1].Similarly we can see that the arrivals of customers into thesystem may also affect the level of congestion. For example,customers impatiencewill affect the arrivals of customers intothe system. A queueing system where the arrival rate and/orservice rate depends on the system size is called adaptivequeueing system.

Sometimes the arrival rate of customers into the systemdepends on the system size instead of a constant rate.Discouraged arrival is one formof state dependence. Here thearrivals get discouraged from joining the queue when moreandmore people are present in the system.We canmodel thiseffect by taking the birth and death coefficients, respectively,as 𝜆𝑛

= 𝜆/(𝑛 + 1), where 𝑛 is the number of customers inthe system and 𝜆 is a positive constant, and 𝜇

𝑛= 𝜇, where

𝜇 is the constant service rate. Thus the arrival rate of thequeueing system is decreased by this discouragement; con-sequently the congestion of the system decreases. This typeof queueing systems is called discouraged arrival queueingsystem.

Hindawi Publishing CorporationAdvances in Operations ResearchVolume 2016, Article ID 2456135, 11 pageshttp://dx.doi.org/10.1155/2016/2456135

2 Advances in Operations Research

Applications of queueing with impatience can be seenin traffic modeling, business and industries, computer com-munication, health sectors, medical sciences, and so forth.Customers with impatience and discouragement have theirown impact on the system performance of standard queueingsystems. It is important to note that customers impatiencehas a very negative impact on the queueing system underinvestigation.

We can see a large number of articles which discussingthe congestion control of queueing systems. For example, see,Haight [2], Ancker Jr. and Gafarian [3, 4], Boots and Tijms[5], Liu and Kulkarni [6], Kc and Terwiesch [7], Wang et al.[8], Kapodistria [9], and Kumar and Sharma [10].

In this paper we attempt to reduce the congestion intwo ways. First we attempt to reduce the congestion bydiscouraging the arrivals of customers from joining thequeue. In the second way we further reduce the congestionby introducing the concept of service switches which isdiscussed in Abdul Rasheed and Manoharan [11].

Discouraged arrival system was studied by manyresearchers. Raynolds [12] presented multiserver queueingmodel with discouragement and obtained equilibriumdistribution of queue length and derived other performancemeasures from it. A finite capacity M/G/1 queueing modelwhere the arrival and the service rates were arbitraryfunctions of the current number of customers in the systemwas studied by Courtois and Georges [13]. Natvig [14]studied the single server birth-death queueing process withstate dependent parameters 𝜆

𝑛= 𝜆/(𝑛 + 1), 𝑛 ≥ 0, and

𝜇𝑛

= 𝜇, 𝑛 ≥ 1. Van Doorn [15] obtained exact expressionsfor transient state probabilities of the birth-death processwith parameters 𝜆

𝑛= 𝜆/(𝑛 + 1), 𝑛 ≥ 0, and 𝜇

𝑛= 𝜇, 𝑛 ≥ 1.

Parthasarathy and Selvaraju [16] obtained the transientsolution to a state dependent birth-death queueing model inwhich potential customers are discouraged by queue length.

Narayanan [17] studied different linear and nonlinearstate dependent Markovian queueing models, in whicharrival rates and/or service rates are nonlinear and theirmodified forms obtain the transient/steady state probabilitydistribution of queue length. Narayanan and Manoharan[18] considered nonlinear state dependent queueing models,in which arrival rates and/or service rates are nonlinear.Narayanan and Manoharan [19] derived the performancemeasures of state dependent queueing models. Ammar et al.[20] studied single server finite capacity Markovian queuewith discouraged arrivals and reneging usingmatrix method.

Abdul Rasheed andManoharan [11] studied a Markovianqueueing system in which the arrival rate is constant andthe service rate depends on the number of customers in thesystem.The server speed is regulated according to the systemsize by introducing the service switches to the model. Theauthors analyzed the system by calculating the performancemeasures such as expected number of customers in thesystem/queue and expected waiting time of customers in thesystem/queue. Some generalizations of the above models arealso presented therein.

A generalization of M/M/𝐶 queueing system with serviceswitches is considered in this paper in which the arrival rate𝜆𝑛and the service rate 𝜇

𝑛are both functions of 𝑛, the number

of customers present in the system. In real life situations 𝜆𝑛

and 𝜇𝑛change whenever 𝑛 changes, so that both arrival and

departure have a bearing on the system state.In many practical queueing systems, when there is a long

queue, it is quite likely that a server will tend to work fasterthan when the queue is small. That is, the service rate 𝜇

𝑛

depends on 𝑛, the number of customers present in the system.Similarly situationsmay occur where customers refuse to jointhe queue because of long waiting by seeing a large numberof customers in the queue.That is, the arrival rate 𝜆

𝑛depends

on 𝑛, the number of customers present in the system. Thesekinds of adaptive queueing systems where the arrival rate andthe service rate depend on the number of customers presentin the system are discussed in this paper.

2. Multiserver Multirate DiscouragedArrival Queueing System

A queue is an indication of congestion which we can beseen in a system or a network consisting of many systems.Congestion arises in many areas and our interest is to controlthe congestion in whatever situation it arises. Abdul Rasheedand Manoharan [11] used the concept of service switches asa tool to control congestion and use multiple servers andmultiple service switches if congestion is very high. Theydiscussed congestion control aspects when the arrival rate isconstant and the service rate depends on the number in thesystem. In this paper we discuss the congestion control usingservice switcheswhenboth the arrival rate and service rate arethe functions of the number of customers in the system. Weconsider a generalized model with 𝐶 servers and two serviceswitches at the point 𝐾

1and 𝐾

2(𝐾2

> 𝐾1) and hence the

system works in three speeds, say, slow, medium, and fast.Here work is performed at the slow rates until there are 𝐾

1

customers in the system, at which point there is a switch to themedium rate and work is performed at the medium rate untilthere are 𝐾

2customers in the system at which point there is

a switch to the fast rate. Here the arrival rate 𝜆𝑛is given as

𝜆𝑛

=𝜆

𝑛 + 1, 𝑛 ≥ 0, (1)

which means the customers will be discouraged from joiningthe queue and the service rate 𝜇

𝑛is given by

𝜇𝑛

=

{{{{{{{

{{{{{{{

{

𝑛𝜇1, 1 ≤ 𝑛 < 𝐶,

𝐶𝜇1, 𝐶 ≤ 𝑛 < 𝐾

1,

𝐶𝜇2, 𝐾1

≤ 𝑛 < 𝐾2,

𝐶𝜇, 𝑛 ≥ 𝐾2,

(2)

where 𝜇1

≤ 𝜇2

< 𝜇.

Advances in Operations Research 3

The steady state probabilities are given by

𝑃𝑛

=

{{{{{{{{{{{{{{{{{

{{{{{{{{{{{{{{{{{

{

𝑟𝑛

1

(𝑛!)2

𝑃0, 0 ≤ 𝑛 < 𝐶,

𝑟𝑛

1

𝑛!𝐶!𝐶𝑛−𝐶𝑃0, 𝐶 ≤ 𝑛 < 𝐾

1,

𝑟𝐾1−1

1𝑟𝑛−𝐾1+1

2

𝑛!𝐶!𝐶𝑛−𝐶𝑃0, 𝐾

1≤ 𝑛 < 𝐾

2,

𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾2+1

𝑛!𝐶!𝐶𝑛−𝐶𝑃0, 𝑛 ≥ 𝐾

2,

(3)

where 𝑟1

= 𝜆/𝜇1, 𝑟2

= 𝜆/𝜇2, and 𝑟 = 𝜆/𝜇.

The idle probability 𝑃0can be obtained as

𝑃0

= [

𝐶−1

∑

𝑛=0

𝑟𝑛

1

(𝑛!)2

+

𝐾1−1

∑

𝑛=𝐶

𝑟𝑛

1

𝑛!𝐶!𝐶𝑛−𝐶+

𝐾2−1

∑

𝑛=𝐾1

𝑟𝐾1−1

1𝑟𝑛−𝐾1+1

2

𝑛!𝐶!𝐶𝑛−𝐶

+

∞

∑

𝑛=𝐾2

𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾2+1

𝑛!𝐶!𝐶𝑛−𝐶]

−1

.

(4)

After a careful manipulation of the infinite series on the righthand side of the above expression and further simplification,we get

𝑃0

= [(𝑟𝐾2−𝐾1

2𝑟1−𝐾2𝑒𝑟/𝐶

𝐶!𝐶−𝐶

+

𝐶−1

∑

𝑛=0

(𝑟𝑛−𝐾1+1

1

(𝑛!)2

−𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾2+1

𝐶!𝐶𝑛−𝐶)

+

𝐾1−1

∑

𝑛=𝐶

(𝑟𝑛−𝐾1+1

1− 𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾2+1

𝑛!𝐶!𝐶𝑛−𝐶)

+

𝐾2−1

∑

𝑛=𝐾1

(𝑟𝑛−𝐾2+1

2− 𝑟𝑛−𝐾2+1

𝑛!𝐶!𝐶𝑛−𝐶) 𝑟𝐾2−𝐾1

2) 𝑟𝐾1−1

1]

−1

.

(5)

The expected queue size (𝐿𝑞) is given as

𝐿𝑞

=

𝐾1−1

∑

𝑛=𝐶+1

(𝑛 − 𝐶) 𝑃𝑛

+

𝐾2−1

∑

𝑛=𝐾1


+

∞

∑

𝑛=𝐾2


= 𝐿𝑞1

+ 𝐿𝑞2

+ 𝐿𝑞3

,

(6)

where

𝐿𝑞1

=

𝐾1−1

∑

𝑛=𝐶+1

𝑟𝑛

1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶−

𝐾1−1

∑

𝑛=𝐶+1

𝑟𝑛

1

𝑛! (𝐶 − 1)!𝐶𝑛−𝐶,

𝐿𝑞2

=

𝐾2−1

∑

𝑛=𝐾1

𝑟𝐾1−1

1𝑟𝑛−𝐾1+1

2

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶𝑃0

−

𝐾2−1

∑

𝑛=𝐾1

𝑟𝐾1−1

1𝑟𝑛−𝐾1+1

2

𝑛! (𝐶 − 1)!𝐶𝑛−𝐶𝑃0,

𝐿𝑞3

=

∞

∑

𝑛=𝐾2

𝑛𝑃𝑛

− 𝐶

∞

∑

𝑛=𝐾2

𝑃𝑛

= 𝐿𝑞31

− 𝐿𝑞32

,

𝐿𝑞31

=𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2𝑟2−𝐾2

𝐶!𝐶1−𝐶𝑃0

∞

∑

𝑛=𝐾2

𝑟𝑛−1

𝐶𝑛−1 (𝑛 − 1)!,

𝐿𝑞31

= [𝑟𝐾1−1

1𝑟𝐾2−𝐾1


𝑃0

𝐶!𝐶1−𝐶

−𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2𝑃0

𝐶!

𝐶

∑

𝑛=1

𝑟𝑛−𝐾2+1

(𝑛 − 1)!𝐶𝑛−𝐶

−𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2𝑃0

𝐶!

𝐾1−1

∑

𝑛=𝐶+1

𝑟𝑛−𝐾2+1

(𝑛 − 1)!𝐶𝑛−𝐶

−𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2𝑃0

𝐶!

𝐾2−1

∑

𝑛=𝐾1

𝑟𝑛−𝐾2+1

(𝑛 − 1)!𝐶𝑛−𝐶] ,

𝐿𝑞32

= 𝐶 [

∞

∑

𝑛=0

𝑃𝑛

−

𝐶

∑

𝑛=1

𝑃𝑛

− 𝑃0

−

𝐾1−1

∑

𝑛=𝐶+1

𝑃𝑛

−

𝐾2−1

∑

𝑛=𝐾1

𝑃𝑛]

= [𝐶 (1 − 𝑃0) − 𝐶

𝐶

∑

𝑛=1

𝑟𝑛

1

(𝑛!)2

𝑃0

− 𝐶

𝐾1−1

∑

𝑛=𝐶+1

𝑟𝑛

1

𝑛!𝐶!𝐶𝑛−𝐶𝑃0

− 𝐶

𝐾2−1

∑

𝑛=𝐾1

𝑟𝐾1−1

1𝑟𝑛−𝐾1+1

2

𝑛!𝐶!𝐶𝑛−𝐶𝑃0] .

(7)

Therefore 𝐿𝑞3becomes

𝐿𝑞3

= [𝑟𝐾1−1

1𝑟𝐾2−𝐾1


𝑃0

𝐶!𝐶1−𝐶

−𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2𝑃0

𝐶!

𝐶

∑

𝑛=1

𝑟𝑛−𝐾2+1

(𝑛 − 1)!𝐶𝑛−𝐶

−𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2𝑃0

𝐶!

𝐾1−1

∑

𝑛=𝐶+1

𝑟𝑛−𝐾2+1

(𝑛 − 1)!𝐶𝑛−𝐶

−𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2𝑃0

𝐶!

𝐾2−1

∑

𝑛=𝐾1

𝑟𝑛−𝐾2+1

(𝑛 − 1)!𝐶𝑛−𝐶− 𝐶 (1 − 𝑃

0)


+ 𝐶

𝐶

∑

𝑛=1

𝑟𝑛

1

(𝑛!)2

𝑃0

+ 𝐶

𝐾1−1

∑

𝑛=𝐶+1

𝑟𝑛

1

𝑛!𝐶!𝐶𝑛−𝐶𝑃0

+ 𝐶

𝐾2−1

∑

𝑛=𝐾1

𝑟𝐾1−1

1𝑟𝑛−𝐾1+1

2

𝑛!𝐶!𝐶𝑛−𝐶𝑃0] .

(8)

After some steps we get expected number of customers in thequeue by using 𝐿

𝑞1, 𝐿𝑞2, and 𝐿

𝑞3as

𝐿𝑞

= [𝑟𝐾1−1

1𝑟𝐾2−𝐾1


𝑃0

𝐶!𝐶1−𝐶

+

𝐶

∑

𝑛=1

(𝐶𝑟𝑛−𝐾1+1

1

(𝑛!)2


2𝑟𝑛−𝐾2+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶) 𝑟𝐾1−1

1𝑃0

+

𝐾1−1

∑

𝑛=𝐶+1

(𝑟𝑛−𝐾1+1

1− 𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾2+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶) 𝑟𝐾1−1

1𝑃0

+

𝐾2−1

∑

𝑛=𝐾1

(𝑟𝑛−𝐾2+1

2− 𝑟𝑛−𝐾2+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶) 𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2𝑃0

− 𝐶 (1 − 𝑃0)] .

(9)

The expected system size (𝐿) is given as

𝐿 =

𝐶

∑

𝑛=0

𝑛𝑃𝑛

+

𝐾1−1

∑

𝑛=𝐶+1

𝑛𝑃𝑛

+

𝐾2−1

∑

𝑛=𝐾1

𝑛𝑃𝑛

+

∞

∑

𝑛=𝐾2

𝑛𝑃𝑛

= [

𝐶

∑

𝑛=0

𝑛𝑃𝑛

+

𝐾1−1

∑

𝑛=𝐶+1


+

𝐾2−1

∑

𝑛=𝐾1


+

∞

∑

𝑛=𝐾2


+ 𝐶 (

𝐾1−1

∑

𝑛=𝐶+1

𝑃𝑛

+

𝐾2−1

∑

𝑛=𝐾1

𝑃𝑛

+

∞

∑

𝑛=𝐾2

𝑃𝑛)] ;

(10)

that is,

𝐿 = 𝐿𝑞

+ 𝑟, (11)

where

𝑟 =

𝐶

∑

𝑛=0

𝑛𝑃𝑛

+ 𝐶

∞

∑

𝑛=𝐶+1

𝑃𝑛,

𝑟 = [

𝐶

∑

𝑛=0

𝑛𝑃𝑛

+ 𝐶

𝐾1−1

∑

𝑛=𝐶+1

𝑃𝑛

+ 𝐶

𝐾2−1

∑

𝑛=𝐾1

𝑃𝑛

+ 𝐶

∞

∑

𝑛=0

𝑃𝑛

− 𝐶

𝐶

∑

𝑛=0

𝑃𝑛

− 𝐶

𝐾1−1

∑

𝑛=𝐶+1

𝑃𝑛

− 𝐶

𝐾2−1

∑

𝑛=𝐾1

𝑃𝑛] = 𝐶 +

𝐶

∑

𝑛=0

(𝑛 − 𝐶)𝑟𝑛

1

(𝑛!)2

𝑃0.

(12)

Hence the expected number of customers in the system (𝐿) isobtained as

𝐿 = [𝑟𝐾1−1

1𝑟𝐾2−𝐾1


𝑃0

𝐶!𝐶1−𝐶

+

𝐶

∑

𝑛=1


1

(𝑛!)2


2𝑟𝑛−𝐾2+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶) 𝑟𝐾1−1

1𝑃0

+

𝐾1−1

∑

𝑛=𝐶+1

(𝑟𝑛−𝐾1+1

1− 𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾2+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶) 𝑟𝐾1−1

1𝑃0

− 𝐶 (1 − 𝑃0)

+

𝐾2−1

∑

𝑛=𝐾1

(𝑟𝑛−𝐾2+1

2− 𝑟𝑛−𝐾2+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶) 𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2𝑃0

+ 𝐶

+

𝐶

∑

𝑛=0

(𝑛 − 𝐶)𝑟𝑛

1

(𝑛!)2

𝑃0] = [

𝑟𝐾2−𝐾1


𝐶!𝐶1−𝐶

+

𝐶

∑

𝑛=1

(𝑟𝑛−𝐾1+1

1

(𝑛 − 1)!𝑛!−

𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾2+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶)

+

𝐾1−1

∑

𝑛=𝐶+1

(𝑟𝑛−𝐾1+1

1− 𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾2+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶)

+

𝐾2−1

∑

𝑛=𝐾1

(𝑟𝑛−𝐾2+1

2− 𝑟𝑛−𝐾2+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶) 𝑟𝐾2−𝐾1

2] 𝑟𝐾1−1

1𝑃0.

(13)

The effective arrival rate (𝜆∗

) can be obtained by the followingsummation schemes:

𝜆∗

=

𝐶−1

∑

𝑛=0

𝜆𝑛𝑃𝑛

+

𝐾1−1

∑

𝑛=𝐶

𝜆𝑛𝑃𝑛

+

𝐾2−1

∑

𝑛=𝐾1

𝜆𝑛𝑃𝑛

+

∞

∑

𝑛=𝐾2

𝜆𝑛𝑃𝑛

= [

𝐶−1

∑

𝑛=0

𝑟𝑛

1

(𝑛 + 1)!𝑛!+

𝐾1−1

∑

𝑛=𝐶

𝑟𝑛

1

(𝑛 + 1)!𝐶!𝐶𝑛−𝐶

+

𝐾2−1

∑

𝑛=𝐾1

𝑟𝐾1−1

1𝑟𝑛−𝐾1+1

2

(𝑛 + 1)!𝐶!𝐶𝑛−𝐶

+𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2𝑟−𝐾2

𝐶!𝐶−𝐶−1(

∞

∑

𝑛=0

𝜌𝑛+1

(𝑛 + 1)!−

𝐶−1

∑

𝑛=0

𝜌𝑛+1

(𝑛 + 1)!

−

𝐾1−1

∑

𝑛=𝐶

𝜌𝑛+1

(𝑛 + 1)!)] 𝜆𝑃

0= [

[

𝑟𝐾2−𝐾1

2𝑟−𝐾2 (𝑒𝑟/𝐶

− 1)

𝐶!𝐶−𝐶−1

+

𝐶−1

∑

𝑛=0

(𝑟𝑛−𝐾1+1

1

(𝑛 + 1)!𝑛!−

𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾2+1

(𝑛 + 1)!𝐶!𝐶𝑛−𝐶)


+

𝐾1−1

∑

𝑛=𝐶

(𝑟𝑛−𝐾1+1

1− 𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾1+1

(𝑛 + 1)!𝐶!𝐶𝑛−𝐶)

+

𝐾2−1

∑

𝑛=𝐾1

(𝑟𝑛−𝐾2+1

2− 𝑟𝑛−𝐾2+1

(𝑛 + 1)!𝐶!𝐶𝑛−𝐶) 𝑟𝐾2−𝐾1

2]

]

𝑟𝐾1−1

1𝜆𝑃0.

(14)

The expected waiting times in the system are

𝑊 = (𝑟𝐾2−𝐾1


𝐶!𝐶1−𝐶

+

𝐶

∑

𝑛=1

(𝑟𝑛−𝐾1+1

1

(𝑛 − 1)!𝑛!−

𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾2+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶)

+

𝐾1−1

∑

𝑛=𝐶+1

(𝑟𝑛−𝐾1+1

1− 𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾2+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶)

+

𝐾2−1

∑

𝑛=𝐾1

(𝑟𝑛−𝐾2+1

2− 𝑟𝑛−𝐾2+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶) 𝑟𝐾2−𝐾1

2)

⋅ ((𝑟𝐾2−𝐾1


− 1)


+

𝐶−1

∑

𝑛=0

(𝑟𝑛−𝐾1+1

1

(𝑛 + 1)!𝑛!−

𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾2+1

(𝑛 + 1)!𝐶!𝐶𝑛−𝐶)

+

𝐾1−1

∑

𝑛=𝐶

(𝑟𝑛−𝐾1+1

1− 𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾1+1

(𝑛 + 1)!𝐶!𝐶𝑛−𝐶)

+

𝐾2−1

∑

𝑛=𝐾1

(𝑟𝑛−𝐾2+1

2− 𝑟𝑛−𝐾2+1

(𝑛 + 1)!𝐶!𝐶𝑛−𝐶) 𝑟𝐾2−𝐾1

2) 𝜆)

−1

.

(15)

Similarly the expected waiting times in the queue are

𝑊𝑞

= ((𝑟𝐾2−𝐾1


𝐶!𝐶1−𝐶

+

𝐶

∑

𝑛=1


1

(𝑛!)2


2𝑟𝑛−𝐾2+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶)

+

𝐾1−1

∑

𝑛=𝐶+1

(𝑟𝑛−𝐾1+1

1− 𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾2+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶)

+

𝐾2−1

∑

𝑛=𝐾1

(𝑟𝑛−𝐾2+1

2− 𝑟𝑛−𝐾2+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶) 𝑟𝐾2−𝐾1

2) 𝑟𝐾1−1

1𝑃0

− 𝐶 (1 − 𝑃0)) ((

𝑟𝐾2−𝐾1


− 1)


+

𝐶−1

∑

𝑛=0

(𝑟𝑛−𝐾1+1

1

(𝑛 + 1)!𝑛!−

𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾2+1

(𝑛 + 1)!𝐶!𝐶𝑛−𝐶)

+

𝐾1−1

∑

𝑛=𝐶

(𝑟𝑛−𝐾1+1

1− 𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾1+1

(𝑛 + 1)!𝐶!𝐶𝑛−𝐶)

+

𝐾2−1

∑

𝑛=𝐾1

(𝑟𝑛−𝐾2+1

2− 𝑟𝑛−𝐾2+1

(𝑛 + 1)!𝐶!𝐶𝑛−𝐶) 𝑟𝐾2−𝐾1

2) 𝑟𝐾1−1

1𝜆𝑃0)

−1

.

(16)

For illustrating the analytical feasibility of the methodsproposed we consider the following hypothetical examplesituation.

Example 1. A young hard worker started a beauty parlour.Customers are taken on a first come first serve basis. Insidethe beauty parlour, sitting facility is available for waitingcustomers and in front of the beauty parlour there is a vastparking area, so no limitation on the number of customerswho can wait for service. But the number of arrivals dependson the number of customers already present in the beautyparlour. If arriving customers see a large number in thesystem he may not join the queue. Since the congestion isvery high the young man appointed one more worker and hedecides to run the beauty parlour at three speeds, say, slow,medium, and fast. At the slow speed, it takes 40 minutes, onthe average; at the medium speed, it takes 30 minutes; and atthe fast speed, it takes 20 minutes to cut the hair with serviceswitches at 5 and 7. That is, up to 4 customers in the systemthe beauty parlour runs at the slow speed. If the number ofcustomers is more than 4 but less than 7, the beauty parlourruns at the medium speed. If the number of customers ismore than 6, the beauty parlour runs at the fast speed. Thatis, 𝐾1

= 5 and 𝐾2

= 7. The interarrival time of customers is35 minutes.

Now we can calculate the measures of effectiveness.If 𝐾1

= 5, 𝐾2

= 7, 𝜆 = 1/35, 𝜇1

= 1/40, 𝜇2

= 1/30,𝜇 = 1/20, and 𝐶 = 2. Then we get 𝜆

∗= 0.0190, 𝑃

0= 0.3935,

𝐿 = 0.795, 𝐿𝑞

= 0.032, 𝑊 = 41.68 minutes, and 𝑊𝑞

= 1.7

minutes.Figure 1 gives the graph of steady state probability of

number of customers in the system.If 𝐾1

= 3 and 𝐾2

= 5 and using the same parameterswe get 𝜆

∗= 0.0190, 𝑃

0= 0.3966, 𝐿 = 0.776, 𝐿

𝑞= 0.022,

𝑊 = 40.46 minutes, and 𝑊𝑞

= 1.18 minutes.If 𝐾1

= 5, 𝐾2

= 7, 𝜆 = 1/35, 𝜇1

= 1/40, 𝜇2

= 1/30,𝜇 = 1/20, and 𝐶 = 1. Then we get 𝜆

∗= 0.0170, 𝑃

0= 0.3195,

𝐿 = 1.134, 𝐿𝑞

= 0.4544, 𝑊 = 66.56 minutes, and 𝑊𝑞

= 26.65

minutes.Figure 2 gives the graph of steady state probability of

number of customers in the system.From this examplewe can observe that waiting time of the

customers decreases if the values of the switch point decreaseand also by increasing the number of servers.

Some special cases of the generalized 𝐶 server model andtwo service switches are discussed in the following section.


Steady state probability of number of customers in the system

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Stea

dy st

ate p

roba

bilit

y

5 10 15 200Number of customers in the system

Figure 1: [𝐶 = 1, 𝜆 = 1/35, 𝜇1

= 1/40, 𝜇2

= 1/30, 𝜇 =

1/20, 𝐾1

= 5, 𝐾2

= 7].

Steady state probability of number of customers in the system

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Stea

dy st

ate p

roba

bilit

y

5 10 15 200Number of customers in the system

Figure 2: [𝐶 = 2, 𝜆 = 1/35, 𝜇1

= 1/40, 𝜇2

= 1/30, 𝜇 =

1/20, 𝐾1

= 5, 𝐾2

= 7].

3. Special Cases

3.1. Model with 𝐶 Servers and One Service Switch. If 𝜇1

= 𝜇2

and hence 𝑟1

= 𝑟2(which means one switch), the model with

𝐶 server and two service switches reduces to the 𝐶 servermodel with one service switch at the point 𝐾 and hence thesystem works in two speeds, say, slow and fast. The followingresults are deduced from the 𝐶 server model and two serviceswitches. The arrival rate 𝜆

𝑛and service rate 𝜇

𝑛are given by

𝜆𝑛

=𝜆

𝑛 + 1, 𝑛 ≥ 0,

𝜇𝑛

=

{{{{

{{{{

{

𝑛𝜇1, 1 ≤ 𝑛 < 𝐶,

𝐶𝜇1, 𝐶 ≤ 𝑛 < 𝐾,

𝐶𝜇, 𝑛 ≥ 𝐾,

(17)

where 𝜇1

< 𝜇 and 𝐾 > 𝐶.


𝑃𝑛

=

{{{{{{{{{{

{{{{{{{{{{

{

𝑟𝑛

1

(𝑛!)2

𝑃0, 0 ≤ 𝑛 < 𝐶,

𝑟𝑛

1

𝑛!𝐶!𝐶𝑛−𝐶𝑃0, 𝐶 ≤ 𝑛 < 𝐾,

𝑟𝐾−1

1𝑟𝑛−𝐾+1

𝑛!𝐶!𝐶𝑛−𝐶𝑃0, 𝑛 ≥ 𝐾.

(18)

The idle probability 𝑃0is obtained as

𝑃0

= [

𝐶−1

∑

𝑛=0

(𝑟𝑛

1

(𝑛!)2

−𝑟𝐾−1

1𝑟𝑛−𝐾+1


+

𝐾−1

∑

𝑛=𝐶

(𝑟𝑛

1

𝑛!𝐶!𝐶𝑛−𝐶−

𝑟𝐾−1

1𝑟𝑛−𝐾+1


+𝑟𝐾−1

1𝑟1−𝐾

𝑒𝑟/𝐶

𝐶!𝐶−𝐶]

−1

.

(19)

The expected number of customers in the queue is

𝐿𝑞

= [𝑟𝐾−1

1𝑟2−𝐾

𝑒𝑟/𝐶

𝑃0

𝐶!𝐶1−𝐶

+

𝐶

∑

𝑛=1

(𝐶𝑟𝑛−𝐾+1

1

(𝑛!)2

−𝑟𝑛−𝐾+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶) 𝑟𝐾−1

1𝑃0

+

𝐾−1

∑

𝑛=𝐶+1

(𝑟𝑛−𝐾+1

1− 𝑟𝑛−𝐾+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶) 𝑟𝐾−1

1𝑃0

− 𝐶 (1 − 𝑃0)] .

(20)

The expected number of customers in the system is

𝐿 = [𝑟𝐾−1

1𝑟2−𝐾

𝑒𝑟/𝐶

𝐶!𝐶1−𝐶

+

𝐶

∑

𝑛=1

(𝑟𝑛−𝐾+1

1

(𝑛 − 1)!𝑛!−

𝑟𝑛−𝐾+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶) 𝑟𝐾−1

1

+

𝐾−1

∑

𝑛=𝐶+1

(𝑟𝑛−𝐾+1

1− 𝑟𝑛−𝐾+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶) 𝑟𝐾−1

1] 𝑃0.

(21)

The effective arrival rate can be calculated as follows:

𝜆∗

=

𝐶−1

∑

𝑛=0

𝜆𝑛𝑃𝑛

+

𝐾−1

∑

𝑛=𝐶

𝜆𝑛𝑃𝑛

+

∞

∑

𝑛=𝐾

𝜆𝑛𝑃𝑛

= [

𝐶−1

∑

𝑛=0

(𝑟𝑛−𝐾+1

1

(𝑛 + 1)!𝑛!−

𝑟𝑛−𝐾+1

𝐶! (𝑛 + 1)!𝐶𝑛−𝐶)

+

𝐾−1

∑

𝑛=𝐶

(𝑟𝑛−𝐾+1

1− 𝑟𝑛−𝐾+1

(𝑛 + 1)!𝐶!𝐶𝑛−𝐶) +

𝑟−𝐾

(𝑒𝑟/𝐶

− 1)

𝐶!𝐶−𝐶−1]

⋅ 𝑟𝐾−1

1𝜆𝑃0.

(22)


The expected waiting time in the system 𝑊 is given by

𝑊 = [

[

(𝑟2−𝐾

𝑒𝑟/𝐶

𝐶!𝐶1−𝐶

+

𝐶

∑

𝑛=1

(𝑟𝑛−𝐾+1

1

(𝑛 − 1)!𝑛!−

𝑟𝑛−𝐾+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶)

+

𝐾−1

∑

𝑛=𝐶+1

(𝑟𝑛−𝐾+1

1− 𝑟𝑛−𝐾+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶))

⋅ ((

𝐶−1

∑

𝑛=0

(𝑟𝑛−𝐾+1

1

(𝑛 + 1)!𝑛!−

𝑟𝑛−𝐾+1

𝐶! (𝑛 + 1)!𝐶𝑛−𝐶)

+

𝐾−1

∑

𝑛=𝐶

(𝑟𝑛−𝐾+1

1− 𝑟𝑛−𝐾+1

(𝑛 + 1)!𝐶!𝐶𝑛−𝐶) +

𝑟−𝐾

(𝑒𝑟/𝐶

− 1)

𝐶!𝐶−𝐶−1)

⋅ 𝜆)

−1

]

]

,

(23)

and expected waiting time in the queue 𝑊𝑞is

𝑊𝑞

= [

[

(𝑟𝐾−1

1𝑟2−𝐾

𝑒𝑟/𝐶

𝑃0

𝐶!𝐶1−𝐶

+

𝐶

∑

𝑛=1

(𝐶𝑟𝑛−𝐾+1

1

(𝑛!)2

−𝑟𝑛−𝐾+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶) 𝑟𝐾−1

1𝑃0

+

𝐾−1

∑

𝑛=𝐶+1

(𝑟𝑛−𝐾+1

1− 𝑟𝑛−𝐾+1

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶) 𝑟𝐾−1

1𝑃0

− 𝐶 (1 − 𝑃0))

⋅ ((

𝐶−1

∑

𝑛=0

(𝑟𝑛−𝐾+1

1

(𝑛 + 1)!𝑛!−

𝑟𝑛−𝐾+1

𝐶! (𝑛 + 1)!𝐶𝑛−𝐶)

+

𝐾−1

∑

𝑛=𝐶

(𝑟𝑛−𝐾+1

1− 𝑟𝑛−𝐾+1

(𝑛 + 1)!𝐶!𝐶𝑛−𝐶) +

𝑟−𝐾

(𝑒𝑟/𝐶

− 1)

𝐶!𝐶−𝐶−1)

⋅ 𝑟𝐾−1

1𝜆𝑃0)

−1

]

]

.

(24)

3.2.Multiple Server AdaptiveQueueing System. If𝜇1

= 𝜇2

= 𝜇

and hence 𝑟1

= 𝑟2

= 𝑟 (which means no switch), the modelwith𝐶 server and two service switches reduces to the𝐶 servermodel with no service switch. The following results can beobtained from the 𝐶 server model and two service switches.The service rate 𝜇

𝑛is given by

𝜇𝑛

={

{

{

𝑛𝜇, 1 ≤ 𝑛 < 𝐶,

𝐶𝜇, 𝑛 ≥ 𝐶.

(25)

The steady state probability of 𝑛 customers in the system is

𝑃𝑛

=

{{{

{{{

{

𝑟𝑛

(𝑛!)2

𝑃0, 0 ≤ 𝑛 < 𝐶,

𝑟𝑛

𝑛!𝐶!𝐶𝑛−𝐶𝑃0, 𝑛 ≥ 𝐶.

(26)

The idle probability can be obtained as

𝑃0

= [

𝐶−1

∑

𝑛=0

𝑟𝑛

(𝑛!)2

+

∞

∑

𝑛=𝐶

𝑟𝑛

𝑛!𝐶!𝐶𝑛−𝐶]

−1

. (27)

The expected number of customers in the system (𝐿) is

𝐿 = [

𝐶

∑

𝑛=1

𝑟𝑛

(𝑛 − 1)! (𝑛!)+

∞

∑

𝑛=𝐶+1

𝑟𝑛

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶] 𝑃0. (28)

The expected number of customers in the queue (𝐿𝑞) is

𝐿𝑞

= [

∞

∑

𝑛=𝐶+1

𝑟𝑛

(𝑛 − 1)!𝐶!𝐶𝑛−𝐶−

∞

∑

𝑛=𝐶+1

𝑟𝑛

𝑛! (𝐶 − 1)!𝐶𝑛−𝐶]

⋅ 𝑃0.

(29)

The effective arrival rate 𝜆∗ can be obtained by

𝜆∗

= [(𝑒𝑟/𝐶

− 1)

𝑟𝐶!𝐶−𝐶−1

−

𝐶−1

∑

𝑛=0

(𝑟𝑛

𝐶! (𝑛 + 1)!𝐶𝑛−𝐶−

𝑟𝑛

𝑛! (𝑛 + 1)!)] 𝜆𝑃

0.

(30)

Waiting times of customers in the system (𝑊) are given as

𝑊 = [(∑𝐶

𝑛=1(𝑟𝑛/ (𝑛 − 1)! (𝑛!)) + ∑

∞

𝑛=𝐶+1(𝑟𝑛/ (𝑛 − 1)!𝐶!𝐶

𝑛−𝐶))

((𝑒𝑟/𝐶 − 1) /𝑟𝐶!𝐶−𝐶−1 − ∑𝐶−1

𝑛=0[𝑟𝑛/𝐶! (𝑛 + 1)!𝐶𝑛−𝐶 − 𝑟𝑛/𝑛! (𝑛 + 1)!]) 𝜆

] . (31)

Waiting times of customers in the queue (𝑊𝑞) are given as

𝑊𝑞

= [(∑∞

𝑛=𝐶+1(𝑟𝑛/ (𝑛 − 1)!𝐶!𝐶

𝑛−𝐶) − ∑∞

𝑛=𝐶+1(𝑟𝑛/𝑛! (𝐶 − 1)!𝐶

𝑛−𝐶))

((𝑒𝑟/𝐶 − 1) /𝑟𝐶!𝐶−𝐶−1 − ∑𝐶−1

𝑛=0[𝑟𝑛/𝐶! (𝑛 + 1)!𝐶𝑛−𝐶 − 𝑟𝑛/𝑛! (𝑛 + 1)!]) 𝜆

] . (32)


3.3. Model with Single Server and Two Service Switches. If𝐶 =

1, the model with 𝐶 server and two service switches reducesto the single server model with two service switches at thepoints 𝐾

1and 𝐾

2(𝐾1

< 𝐾2). The following results can be

obtained from the multiserver model with two switches. Theservice rate 𝜇

𝑛is given by

𝜇𝑛

=

{{{{

{{{{

{

𝜇1, 1 ≤ 𝑛 < 𝐾

1,

𝜇2, 𝐾1

≤ 𝑛 < 𝐾2,

𝜇, 𝑛 ≥ 𝐾2.

(33)

The steady state probability of 𝑛 customers in the system is

𝑃𝑛

=

{{{{{{{{{{{

{{{{{{{{{{{

{

𝑟𝑛

1

𝑛!𝑃0, 0 ≤ 𝑛 < 𝐾

1,

𝑟𝐾1−1

1𝑟𝑛−𝐾1+1

2

𝑛!𝑃0, 𝐾

1≤ 𝑛 < 𝐾

2,

𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾2+1

𝑛!𝑃0, 𝑛 ≥ 𝐾

2.

(34)

We have the idle probability

𝑃0

= [

[

𝐾1−1

∑

𝑛=0

(𝑟𝐾1−1

1(𝑟𝑛−𝐾1+1

1− 𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾2+1

)

𝑛!)

+

𝐾2−1

∑

𝑛=𝐾1

(𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2(𝑟𝑛−𝐾2+1

2− 𝑟𝑛−𝐾2+1

)

𝑛!)

+ 𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2𝑟1−𝐾2𝑒𝑟]

]

−1

.

(35)

The expected number of customers in the system (𝐿) is givenby

𝐿 = [

[

𝐾1−1

∑

𝑛=1

(𝑟𝐾1−1

1(𝑟𝑛−𝐾1+1

1− 𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾2+1

)

(𝑛 − 1)!)

+

𝐾2−1

∑

𝑛=𝐾1

(𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2(𝑟𝑛−𝐾2+1

2− 𝑟𝑛−𝐾2+1

)

(𝑛 − 1)!)

+ 𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2𝑟2−𝐾2𝑒𝑟]

]

𝑃0.

(36)

The expected number of customers in the queue (𝐿𝑞) is given

by

𝐿𝑞

= [

[

𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2𝑟2−𝐾2𝑒𝑟𝑃0

+

𝐾1−1

∑

𝑛=1

((𝑟𝑛−𝐾1+1

1− 𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾2+1

)

(𝑛 − 1)!) 𝑟𝐾1−1

1𝑃0

+

𝐾2−1

∑

𝑛=𝐾1

((𝑟𝑛−𝐾2+1

2− 𝑟𝑛−𝐾2+1

)

(𝑛 − 1)!) 𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2𝑃0

− (1 − 𝑃0)]

]

.

(37)

We can see that 𝐿𝑞

= 𝐿 − (1 − 𝑃0) from the above equation.

The effective arrival rate now can be obtained as

𝜆∗

= [𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2𝑟−𝐾2 (𝑒𝑟

− 1)

+

𝐾1−1

∑

𝑛=0

(𝑟𝑛−𝐾1+1

1− 𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾1+1

(𝑛 + 1)!) 𝑟𝐾1−1

1

+

𝐾2−1

∑

𝑛=𝐾1

(𝑟𝑛−𝐾2+1

2− 𝑟𝑛−𝐾2+1

(𝑛 + 1)!) 𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2] 𝜆𝑃0.

(38)

The expected waiting time in the system is

𝑊 = [

[

(𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2𝑟2−𝐾2𝑒𝑟

+

𝐾1−1

∑

𝑛=1

[

[

𝑟𝐾1−1

1(𝑟𝑛−𝐾1+1

1− 𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾2+1

)

(𝑛 − 1)!

]

]

+

𝐾2−1

∑

𝑛=𝐾1

[

[

𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2(𝑟𝑛−𝐾2+1

2− 𝑟𝑛−𝐾2+1

)

(𝑛 − 1)!

]

]

)

⋅ ((𝑟𝐾1−1

1𝑟𝐾2−𝐾1


− 1)

+

𝐾1−1

∑

𝑛=0

(𝑟𝑛−𝐾1+1

1− 𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾1+1

(𝑛 + 1)!) 𝑟𝐾1−1

1

+

𝐾2−1

∑

𝑛=𝐾1

(𝑟𝑛−𝐾2+1

2− 𝑟𝑛−𝐾2+1

(𝑛 + 1)!) 𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2) 𝜆)

−1

]

]

.

(39)


The expected waiting time in the queue is

𝑊𝑞

= [

[

(𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2𝑟2−𝐾2𝑒𝑟𝑃0

+

𝐾1−1

∑

𝑛=1

((𝑟𝑛−𝐾1+1

1− 𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾2+1

)

(𝑛 − 1)!) 𝑟𝐾1−1

1𝑃0

+

𝐾2−1

∑

𝑛=𝐾1

((𝑟𝑛−𝐾2+1

2− 𝑟𝑛−𝐾2+1

)

(𝑛 − 1)!) 𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2𝑃0

− (1

− 𝑃0)) ((𝑟

𝐾1−1

1𝑟𝐾2−𝐾1


− 1)

+

𝐾1−1

∑

𝑛=0

(𝑟𝑛−𝐾1+1

1− 𝑟𝐾2−𝐾1

2𝑟𝑛−𝐾1+1

(𝑛 + 1)!) 𝑟𝐾1−1

1

+

𝐾2−1

∑

𝑛=𝐾1

(𝑟𝑛−𝐾2+1

2− 𝑟𝑛−𝐾2+1

(𝑛 + 1)!) 𝑟𝐾1−1

1𝑟𝐾2−𝐾1

2)

⋅ 𝜆𝑃0)

−1

]

]

.

(40)

Also we can establish the relationship 𝑊 = 𝑊𝑞

+ (1 − 𝑃0)/𝜆∗,

so the expected service time is (1 − 𝑃0)/𝜆∗

.

3.4. Model with Single Server and One Service Switch. If 𝐶 =

1, 𝜇1

= 𝜇, and hence 𝑟1

= 𝑟, the model with 𝐶 server andtwo service switches reduces to the single server model withone service switch at the point 𝐾. We obtained the followingresults from the 𝐶 server two switch models. The service rate𝜇𝑛is given as

𝜇𝑛

={

{

{

𝜇1, 1 ≤ 𝑛 < 𝐾,

𝜇, 𝑛 ≥ 𝐾.

(41)


𝑃𝑛

=

{{{{{

{{{{{

{

𝑟𝑛

1

𝑛!𝑃0, 0 ≤ 𝑛 < 𝐾,

𝑟𝐾−1

1𝑟𝑛−𝐾+1

𝑛!𝑃0, 𝑛 ≥ 𝐾.

(42)

The idle probability can be obtained from the result∑∞

𝑛=0𝑃𝑛

=

1, as

𝑃0

= [𝑟𝐾−1

1(𝑟1−𝐾

𝑒𝑟

−

𝐾−1

∑

𝑛=0

(𝑟𝑛−𝐾+1

− 𝑟𝑛−𝐾+1

1)

𝑛!)]

−1

. (43)

The expected number of customers in the system (𝐿) is

𝐿 = [𝑟𝐾−1

1𝑟2−𝐾

𝑒𝑟

+

𝐾−1

∑

𝑛=1

𝑟𝐾−1

1(

𝑟𝑛−𝐾+1

1− 𝑟𝑛−𝐾+1

(𝑛 − 1)!)] 𝑃0. (44)

The expected number of customers in the queue (𝐿𝑞) is

𝐿𝑞

= [(𝑒𝑟𝑟2−𝐾

− (𝑟2−𝐾

− 𝑟2−𝐾

1)

−

𝐾−1

∑

𝑛=2

(𝑟𝑛−𝐾+1

− 𝑟𝑛−𝐾+1

1)

(𝑛 − 1)!) 𝑟𝐾−1

1𝑃0

− (1 − 𝑃0)] .

(45)

The effective arrival rate 𝜆∗ can be obtained as

𝜆∗

= [𝑟𝐾−1

1𝑟−𝐾

(𝑒𝑟

− 1)

+

𝐾−1

∑

𝑛=0

(𝑟𝑛−𝐾+1

1− 𝑟𝑛−𝐾+1

(𝑛 + 1)!) 𝑟𝐾−1

1] 𝜆𝑃0.

(46)

The expected waiting time in the system can be obtained as

𝑊 = [(𝑟𝐾−1

1𝑟2−𝐾

𝑒𝑟

+ ∑𝐾−1

𝑛=1𝑟𝐾−1

1((𝑟𝑛−𝐾+1

1− 𝑟𝑛−𝐾+1

) / (𝑛 − 1)!))

(𝑟𝐾−1

1𝑟−𝐾 (𝑒𝑟 − 1) + ∑

𝐾−1

𝑛=0((𝑟𝑛−𝐾+1

1− 𝑟𝑛−𝐾+1) / (𝑛 + 1)!) 𝑟

𝐾−1

1) 𝜆

] . (47)

Similarly the expected waiting time of customers in the queuecan be obtained as

𝑊𝑞

= [((𝑒𝑟𝑟2−𝐾

− (𝑟2−𝐾

− 𝑟2−𝐾

1) − ∑𝐾−1

𝑛=2((𝑟𝑛−𝐾+1

− 𝑟𝑛−𝐾+1

1) / (𝑛 − 1)!)) 𝑟

𝐾−1

1𝑃0

− (1 − 𝑃0))

(𝑟𝐾−1

1𝑟−𝐾 (𝑒𝑟 − 1) + ∑

𝐾−1

𝑛=0((𝑟𝑛−𝐾+1

1− 𝑟𝑛−𝐾+1) / (𝑛 + 1)!) 𝑟

𝐾−1

1) 𝜆𝑃0

] . (48)


3.5. Single Server Adaptive Queueing System. If 𝐶 = 1, 𝜇1

=

𝜇2

= 𝜇, and hence 𝑟1

= 𝑟2

= 𝑟, the model with 𝐶 serverand two service switches reduce to the single server modelwith no service switch and hence the following results can beobtained from the 𝐶 server model and two service switches.The service rate 𝜇

𝑛is given by 𝜇

𝑛= 𝜇, 𝑛 ≥ 1.

The steady state probability of 𝑛 customers in the systemis

𝑃𝑛

=𝑟𝑛

𝑛!𝑃0, (49)

where

𝑃0

= 𝑒−𝑟

. (50)

Hence steady state probability of 𝑛 customers in the systembecomes

𝑃𝑛

=𝑒−𝑟

𝑟𝑛

𝑛!, (51)

which is a Poisson distribution with parameter 𝑟 = 𝜆/𝜇.The expected number of customers in the system (𝐿) is

𝐿 =

∞

∑

𝑛=1

𝑛𝑃𝑛

= 𝑟, (52)

since 𝑃𝑛follows Poisson distribution.

Similarly the expected number of customers in the queue(𝐿𝑞) is

𝐿𝑞

=

∞

∑

𝑛=1

(𝑛 − 1) 𝑃𝑛

= 𝑒−𝑟

+ 𝑟 − 1. (53)

Expected waiting times of customers in the system (𝑊) aregiven as

𝑊 =𝑟

𝜇 (1 − 𝑒−𝑟). (54)

Expected waiting times of customers in the queue (𝑊𝑞) are

given as

𝑊𝑞

=𝑒−𝑟

+ 𝑟 − 1

𝜇 (1 − 𝑒−𝑟). (55)

Sensitivity Analysis. Now we investigate the nature of theexpected number of customers in the system and expectedwaiting time of customers in the system on the basis ofthe various values of the switch point 𝐾 by the followingsensitivity analysis.

FromTable 1 we can observe that as the value of the switchpoint 𝐾 increases, the results of the single server discouragedarrival model with one service switch tending to the results ofthe single server discouraged arrival model with no serviceswitch. That is, if 𝐾 ≥ 9, 𝑃

0= 0.3189 which is the 𝑃

0of

the single server discouraged arrival model with no serviceswitch. Similarly if 𝐾 ≥ 11, 𝐿 = 1.142857 which is the 𝐿 ofthe single server discouraged arrival model with no serviceswitch and if 𝐾 ≥ 12, 𝑊 = 67.118963 which is the 𝑊 ofthe single server discouraged arrival model with no serviceswitch.Hencewe conclude that there is no effect by the switchpoint if its value increases.

Table 1: [𝜆 = 1/35, 𝜇1

= 1/40, 𝜇 = 1/30, 𝐶 = 1].

𝐾 𝑃0

Expected number inthe system (𝐿)

Expected waiting timein the system (𝑊)

2 0.356 0.958 53.054 0.3217 1.114 65.049 0.3189 1.142850 67.11848611 0.3189 1.142857 67.11895712 0.3189 1.142857 67.11896325 0.3189 1.142857 67.11896350 0.3189 1.142857 67.118963100 0.3189 1.142857 67.118963

4. Summary and Concluding Remarks

In this paper we study a discouraged arrival Marko-vian queueing systems. To this system we introduce self-regulatory servers and analyzed the model by deriving steadystate characteristics. A generalized multiple server discour-aged arrival model with two service switches are discussed.By introducing service switches we could speed up the servicebased on the switch point if the number of arrivals increases.That is, the speed of the server can be slow, medium, and fast.Thus the congestion of customers can be reduced by the twofeatures mentioned above. The steady state probabilities andall the performance measures such as expected number ofcustomers in the queue/system and expected waiting time ofcustomers in the queue/system are derived.

From this general model we derived𝐶 server discouragedarrival model with one service switch, single server discour-aged arrival model with two service switches, single serverdiscouraged arrival model with one service switch, multipleserver adaptive queueing system, and single server adaptivequeueing system as a special case.

Numerical illustration of the model is given. A Matlabprogram was developed to help the numerical illustration. Asensitivity analysis is conducted to obtain the optimum valueof the switch point.

From the numerical illustration, we can observe that asthe value of the switch points decreases the waiting time ofcustomers also decreases.

For implementing the above models in real life situationswe need to include the cost of the various service rate of thesystem. An optimization problem can be a future work of themodel by finding the optimal choice of the service switchesand the number of servers by considering the costs involvedin the queueing system.

Competing Interests

The authors declare that they have no competing interests.

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