transient analysis of a two-heterogeneous servers queue with...

TOP (2017) 25:179–205DOI 10.1007/s11750-016-0428-x

ORIGINAL PAPER

Transient analysis of a two-heterogeneous servers queuewith system disaster, server repair and customers’impatience

R. Sudhesh1 · P. Savitha1 · S. Dharmaraja2

Received: 23 March 2016 / Accepted: 23 June 2016 / Published online: 7 July 2016© Sociedad de Estadística e Investigación Operativa 2016

Abstract A two-heterogeneous servers queue with system disaster, server failure andrepair is considered. In addition, the customers become impatient when the system isdown. The customers arrive according to a Poisson process and service time followsexponential distribution. Each customer requires exactly one server for its serviceand the customers select the servers on fastest server first basis. Explicit expressionsare derived for the time-dependent system size probabilities in terms of the modifiedBessel function, by employing the generating function along with continued fractionand the identity of the confluent hypergeometric function. Further, the steady-stateprobabilities of the number of customers in the system are deduced and finally someimportant performance measures are obtained.

Keywords Heterogeneous servers · System disaster · Server repair · Customers’impatience · Confluent hypergeometric function · Moments · Steady-stateprobabilities · Performance measures

Mathematics Subject Classification 60K25

B R. [email protected]

P. [email protected]

S. [email protected]

1 Department of Mathematics, Bharathidasan Institute of Technology (BIT) Campus, AnnaUniversity, Tiruchirappalli 620024, India

2 Department of Mathematics, Indian Institute of Technology Delhi, New Delhi 110016, India

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180 R. Sudhesh et al.

1 Introduction

Multi-server queueing systems has gained a tremendous importance these days insolving various problems in production management, traffic control, computer andcommunication networks and in supply chain management. Numerous researchershave considered multi-server queueing models in different scenarios. Most of thequeueing systems dealt in literature are analyzed in steady-state. However, thesteady-state measures do not reveal the complete picture of the system behaviour,because they ignore the transient and start-up effects. In many potential applica-tions, steady-state measures of the system performance simply do not make sensebecause the analyst needs to know how the system will operate up to some timeinstant t .

Heterogeneity of service is a major aspect of many real multi-server queueing situ-ations such as banks, hospitals, telecommunication networks, manufacturing systemsand several business organisations. The heterogeneous servicemechanisms are invalu-able scheduling methods that allow customers to receive different quality and speedof service. The role of quality and service performance are crucial aspects in customerperceptions and firms must dedicate special attention to them when designing andimplementing their operations. For this reason, the queues with heterogeneous servershave received considerable attention in the literature (Ammar 2014a, b).

In Lippolt et al. (2003) the authors analyzed a single stage production systemwith two-heterogeneous machines in steady-state. They compared the homogeneousmachine centre and heterogeneous machine centre using Markov chain analyzer andconcluded that the performance of the production system can be well approximatedby a heterogeneous system (machines with differing service speed).

In Ke et al. (2015) the authors considered a machine repair problem withqueue-dependent heterogeneous repairmen. They derived the stationary probabilitydistribution of the number of failed machines in the system and implemented thequasi-Newton method to determine the optimal values of threshold for increasing thetotal service rate where the service rate is assumed adjustable.

Moreover the adoption of the first come first served (FCFS) discipline may notbe realistic in modeling service systems with heterogeneous structures. For instance,if there are two-heterogeneous servers in banks who provide services with varyingspeeds then the customers might prefer to choose the fastest server for service. Onthe other hand, if one chooses the slowest server randomly then the customers whoentered the system after him/her may depart earlier by obtaining service from a fastestserver with a faster service rate. Apparently in this case, the FCFS queue discipline isviolated due to heterogeneity in service speed of the servers.

Trivedi (2002) presented the two-server heterogeneous system where one server isfaster than the other andnoqueue is allowed in front of the slower server.He restricts hisanalysis to steady-statemeasures. For the samemodel,Dharmaraja (2000) obtained theexact time-dependent system size probabilities using generating function. Kumar et al.(2007) studied an M/M/2 queue with heterogeneous servers subject to catastrophesand derived the transient solution of the model.

Recently Dharmaraja and Kumar (2015) derived the time-dependent solutions ofthe heterogeneousmultiple server queueing systemwith catastrophes using generating

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Transient analysis of a two-heterogeneous servers queue... 181

function. There are c-heterogeneous servers in the systemwhere the servers are orderedin decreasing service speed. The customers select the servers on fastest server first(FSF) basis. This idea motivates us to model the system with two-heterogeneousservers on FSF basis.

Real life queueing systems such as in telecommunications, disaster may occurleading to the instantaneous departure of all customers from the system. A disaster isalso called a catastrophe, mass exodus, or queue flushing (Chen and Renshaw 1997).A disaster can be treated as a server reset or unplug which breaks down the server andcauses all the customers in the system to be lost. The loss of customers due to disastrousbreakdowns (also referred to as a kind of negative arrivals) was first introduced byGelenbe (1991).

Queueing models with disaster can be used to analyze system breakdowns due toa reset order or computer networks with virus infections. There is a growing interestin the analysis of queueing models subject to system disaster and repairable server;see Ammar (2014b), Kumar (2008) and Sudhesh et al. (2016). Sudhesh et al. (2016)analyzed a single server queueing system with N -Policy and disastrous breakdownand obtained the transient state and steady-state probabilities in closed form.

Most of the works in the literature are carried out based on the assumption thatdisaster (catastrophe) occurs onlywhen the system is not empty.However, this situationis unsuitable in many real-life systems. In various practical situations disaster mayoccur even when the system is empty. This motivates us to consider a system withdisastrous breakdowns when the system is both idle and busy.

Customer impatience is a very important phenomenon in queueing models. Queue-ing models in which customers may balk, or abandon the system before receivingservice arise in many diverse application domains, where customers wait for servicefor a limited time only and leave the system if service has not begin within that time.

In real-time telecommunication systems the subscribers give up due to impatiencebefore the requested connection is established.

Queueing models in which customers leave without getting service either due tomass exodus or due to customers’ impatience have been studied in the literature.Kumar(2007) studied a single server queueing system with catastrophes, server failures andrepairs. They analyzed the transient behaviour of themodel andperform the availabilityand reliability analysis of the system. Yechiali (2007) investigated the single server,homogeneous multi-server and infinite server queueing systems with system disasterand customer impatience and obtained the stationary probabilities of the system. Forthe single server model of Yechiali (2007), Sudhesh (2010) derived the transient-state probability distribution of the system and obtained an explicit expression usingcontinued fraction and generating function. Ammar (2014a) obtained the transientsolution of a two-heterogeneous servers queue with impatient behaviour subject tobalking and reneging.

Recently, Ammar (2015) analyzed a single server queue with impatient customersand multiple vacations, where customers became impatient only due to an absenteeof servers upon arrival.

Ammar (2014b) expands the model described in Dharmaraja (2000) with catastro-phes, server failures and repairs and obtained an exact time-dependent solution.

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However, no work has been found in the literature which studies queueing sys-tems with heterogeneous service, system disaster and server repair taking together theeffect of customers’ impatience. Based on this observation, we have investigated thetransient system size probabilities for the two-heterogeneous server queueing systemsubject to system disaster, server failures, repairs and customers’ impatience by defin-ing a suitable probability generating function, continued fractions (CFs) and identitiesrelated to CFs and confluent hypergeometric function.

Themotivation of thismodel also has an application to the dynamic routing problemin computer systems or communication networks. Messages will arrive at the bufferthat have to be routed over one of several communication lines. The communicationlines are heterogeneous servers which send messages of possibly different speeds.Systemdisastermay occur in the form of virus infectionswhich causes the server downand instantaneously all the messages to be lost. A repair process starts immediatelyand the newly arriving messages are queued in the buffer. If the time taken to transmitthe message is too long the sender may give up and leave the system. Our goal is tominimize the overall delay of the message.

In Armony and Ward (2010) the authors formulated an optimization problem fora call center with heterogeneous agent pools to minimize expected customer waitingtime. They proposed a threshold routing policy which is asymptotically optimal thanthe common routing policy used in call centres.

Recently the authors Ibrahima et al. (2016) developed stochastic models and carriedout a large-scale data-based investigation of service times in a call center with manyheterogeneous agents and multiple call types.

The outline of this paper is given as follows: In Sect. 2, the mathematical model andthe system of differential-difference equations of probabilities governing this modelare presented. In Sect. 3, the time-dependent state probabilities for the number inthe system using generating function, Bessel function and confluent hypergeometricfunction is obtained. InSect. 4, the time-dependentmoments of the system is presented.In Sect. 5, the steady-state probabilities of the system size is deduced. Finally inSect. 6, some important performance measures of the system are presented and thenthe concluding remarks are summarized in Sect. 7.

2 Model description

We consider a single stage production system with two-heterogeneous machines(servers) queueing system subject to disastrous breakdowns, machine repair and job’simpatience. Jobs (Production Orders) arrive according to a Poisson process with rateλ. Service time is exponentially distributed where the two machines namely fastermachine and slower machine provide heterogeneous service with different servicerates μ1 and μ2 such that μ1 > μ2. The machines have a common buffer with jobswaiting for service. Each job needs only one machine for service and the jobs selectthe servers on fastest server first (FSF) basis.

When the system is idle or busy, disaster occurs according to a Poisson process ofrate γ . Whenever a disaster occurs at the system, all present jobs (both waiting andserved) are flushed out from the system and both the machines become inactivated. A

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Fig. 1 Pictorial representation of the model

repair process then starts immediately and the repair time of the system is exponentiallydistributed with mean η−1. When the system is down, inoperative, and machines areundergoing a repair process, new arrival jobs become impatient. Each individual jobupon arrival, activates a random-duration impatience timer which is exponentiallydistributed with parameter ξ . If the timer expires before the machine gets repaired,the job abandons the queue and never return. The diagrammatic representation of themodel is shown in Fig. 1.

Let {(X (t),Y (t)), t ≥ 0} be a two-dimensional continuous-time Markov chain,where X (t) denote the number of customers in the system at time t and Y (t) representsthe state of the system at time t with state space S = {(n, j) : n = 0, 1, 2, . . . ; j =0, 1, 2, 3, 4}.

The state (0, 0) represents that the system is empty and servers are ready to servecustomers, (1, 1) represents that there is one customer in the system and served byfaster server and (1, 2) represents that there is one customer in the system and servedby slower server.

If Y (t) = 3, the system is functioning and both the servers are serving customers,whereas if Y (t) = 4, the system is down and undergoing a repair process.

The state (n, 3), n = 2, 3, 4, . . . , represents that there are n customers in thesystem when the system is in working state and the state (n, 4), n = 0, 1, 2, . . . ,represents that there are n customers in the system when the system is in failure state.

Let Pnj (t) denote the time-dependent system size probabilities where there are ncustomers in the system at time t and j takes values 0, 1, 2, 3 and 4. Mathematically,

Pnj (t) = P[X (t) = n,Y (t) = j], n = 0, 1, 2, . . . ; j = 0, 1, 2, 3, 4

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2.1 The service discipline

The following four cases constitute the service discipline while the customer arrivesinto the system.

1. Both servers are idle: the arriving customer chooses faster server.2. Faster server is busy and slower server is idle: The arriving customer will take

service from slower server.3. Faster server is idle and slower server is busy: The arriving customer prefers faster

server.4. Both servers are busy: The customer will join the queue and wait to take service

from either faster server or slower server based on who completed the service first.This schedule is renewed each time when there are three or more customers in thesystem.

2.2 Governing equations

With the underlying assumptions, the behaviour of resulting system is described by aset of Kolmogorov forward equations which can be written as follows:

P′00(t) = − (λ + γ )P00(t) + μ1P11(t) + μ2P12(t) + ηP04(t), (2.1)

P′11(t) = − (λ + μ1 + γ )P11(t) + λP00(t) + μ2P23(t) + ηP14(t), (2.2)

P′12(t) = − (λ + μ2 + γ )P12(t) + μ1P23(t), (2.3)

P′23(t) = − (λ + μ + γ )P23(t) + λP11(t) + λP12(t) + μP33(t) + ηP24(t), (2.4)

P′n3(t) = − (λ + μ + γ )Pn3(t) + λP(n−1)3(t) + μP(n+1)3(t) + ηPn4(t), n ≥ 3,

(2.5)

P′04(t) = − (λ + η)P04(t) + ξ P14(t) + γ

[1 −

∞∑n=0

Pn4(t)

], (2.6)

P′n4(t) = − (λ + η + nξ)Pn4(t) + λP(n−1)4(t) + (n + 1)ξ P(n+1)4(t), n ≥ 1,

(2.7)

where μ = μ1 + μ2.We assume that initially there are no customers in the system. i.e., P00(0) = 1.

3 Transient behaviour

In this section, the time-dependent system size probabilities can be derived usinggenerating function, Laplace transforms and continued fractions.Define the probability generating function

G(t, z) = R(t) +∞∑n=0

P(n+3)3(t)zn+1, (3.1)

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whereR(t) = P00(t) + P11(t) + P12(t) + P23(t)

with initial condition G(0, z) = 1.The system of Eqs. (2.1)–(2.5) yield the following partial differential equation

∂G

∂t= −γ R(t) + η

[P04(t) + P14(t) + P24(t) +

∞∑n=1

P(n+2)4(t)zn]

+ λ(z − 1)P23(t) +(

λz − (λ + μ + γ ) + μz

)(G(t, z) − R(t)). (3.2)

Integrating, the above equation, we get

G(t, z)=λ(z − 1)∫ t0exp{−(λ+μ+γ )(t−y)} exp

{(λz+ μ

z

)(t−y)

}P23(y)dy

+[(λ + μ) −

(λz + μ

z

)]∫ t0exp {−(λ + μ + γ )(t − y)}

× exp{(

λz + μz

)(t − y)

}R(y)dy

+ η∫ t0exp {−(λ + μ + γ )(t − y)}

× exp{(

λz + μz

)(t − y)

}[P04(y) + P14(y) + P24(y)

+∞∑n=1

P(n+2)4(y)zn]dy + exp {−(λ + μ + γ )(t)} exp

{(λz + μ

z

)(t)

}.

(3.3)

If we assume that α = 2√λμ and β =√

λμthen

exp

{(λz + μ

z

)(t − y)

}=

∞∑n=−∞

(βz)n In[α(t − y)],

where In(.) = In[α(t − y)] is the modified Bessel function of first kind.

3.1 Evaluation of P(n+2)3(t), n ≥ 1

Comparing the coefficients of zn on both sides of Eq. (3.3), for n = 1, 2, 3, . . . , weobtain

β−n P(n+2)3(t) =∫ t0exp{−(λ + μ + γ )(t − y)}

[λβ−1 In−1(.) − λIn(.)

]P23(y)dy

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+∫ t0exp{−(λ + μ + γ )(t − y)}

×[(λ + μ)In(.) − λβ−1 In−1(.) − μβ In+1(.)

]R(y)dy

+ η∫ t0exp{−(λ+μ+γ )(t−y)}In(.)[P04(y)+P14(y)+P24(y)]dy

+ η∫ t0exp {−(λ+μ+γ )(t − y)}

[ ∞∑i=1

P(i+2)4(y)β−i In−i (.)]dy

+ exp {−(λ + μ + γ )(t)} In(αt). (3.4)

The above Eq. (3.4) holds for n = −1,−2,−3, . . . with left hand side replaced byzero. Using Bessel property I−n(.) = In(.) for n = 1, 2, 3, . . . , we get

0 =∫ t0exp{−(λ + μ + γ )(t − y)}

[λβ−1 In+1(.) − λIn(.)

]P23(y)dy

+∫ t0exp{−(λ + μ + γ )(t − y)}

×[(λ + μ)In(.) − λβ−1 In+1(.) − μβ In−1(.)

]R(y)dy

+ η∫ t0exp{−(λ + μ + γ )(t − y)}In(.)[P04(y) + P14(y) + P24(y)]dy

+ η∫ t0exp{−(λ + μ + γ )(t − y)}

∞∑i=1

P(i+2)4(y)β−i In+i (.)]dy

+ exp{−(λ + μ + γ )(t)}In(αt). (3.5)

Subtracting Eq. (3.5) from Eq. (3.4), for n = 1, 2, 3, . . . , we obtain

P(n+2)3(t) = nβn∫ t0exp{−(λ + μ + γ )(t − y)} In(.)

(t − y) P23(y)dy

+ η∫ t0exp{−(λ + μ + γ )(t − y)}

∞∑i=1

βn−i P(i+2)4(y)

×[In−i (.) − I(n+i)(.)]dy. (3.6)

3.2 Evaluation of P23(t)

We rewrite the system of Eqs. (2.1)–(2.3) in the following matrix form:

dH(t)

dt= BH(t) + ηP04(t)e1 + (μ2P23(t) + ηP14(t)) e2 + μ1P23(t)e3, (3.7)

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where

H(t) = (P00(t), P11(t), P12(t))T ,

B =⎡⎣−(λ + γ ) μ1 μ2λ −(λ + μ1 + γ ) 0

0 0 −(λ + μ2 + γ )

⎤⎦ ,

e1 = (1, 0, 0)T , e2 = (0, 1, 0)T and e3 = (0, 0, 1)T .Let f̂ (s) denotes the Laplace transform of f (t). By taking Laplace transform on Eq.(3.7), we get

Ĥ(s)=(s I − B)−1[H(0)+η P̂04(s)e1+

(μ2 P̂23(s)+η P̂14(s)

)e2+μ1 P̂23(s)e3

](3.8)

with H(0) = (1, 0, 0)T . To find P̂23(s), we have

R̂(s) = eT Ĥ(s) + P̂23(s), (3.9)

where e = (1, 1, 1)T . Comparing the constant term in either side of Eq. (3.3) andusing Bessel property, we obtain

R(t) =∫ t0exp{−(λ + μ + γ )(t − y)}

[λβ−1 I1(.) − λI0(.)

]P23(y)dy

+∫ t0exp{−(λ + μ + γ )(t − y)}

×[(λ + μ)I0(.) − λβ−1 I1(.) − μβ I1(.)

]R(y)dy

+ η∫ t0exp{−(λ + μ + γ )(t − y)}I0(.)[P04(y) + P14(y) + P24(y)]dy

+ η∫ t0exp{−(λ + μ + γ )(t − y)}

[ ∞∑i=1

P(i+2)4(y)β−i Ii (.)]dy

+ exp{−(λ + μ + γ )(t)}I0(αt). (3.10)

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Taking Laplace transforms, after some simplification the above equation yield

R̂(s)(s + γ ) = 12P̂23(s)

[w −

√w2 − α2 − 2λ

]+ η[P̂04(s) + P̂14(s) + P̂24(s)

]

+ η∞∑i=1

P̂(i+2)4(s)(

w − √w2 − α22λ

)i+ 1, (3.11)

where w = s + λ + μ + γ .Using Eqs. (3.8) and (3.9) in Eq. (3.11) and simplifying we get,

P̂23(s)=[s+γ − 1

2[w−

√w2 − α2 − 2λ]+(s+γ )eT (s I − B)−1(μ2e2+μ1e3)

]−1

×[1+η[P̂04(s)+ P̂14(s)+ P̂24(s)]+η

∞∑i=1

P̂(i+2)4(s)(w − √w2 − α2

2λ

)i

−eT (s I − B)−1(s + γ )[(1 + P̂04(s)η)e1 + P̂14(s)ηe2]]. (3.12)

Let (s I − B)−1 = (b∗i j (s))3×3 which is

1

|D(s)|

⎡⎣b1(s)b2(s) μ1b2(s) μ2b1(s)λb2(s) c(s)b2(s) λμ2

0 0 c(s)b1(s) − λμ1

⎤⎦ , (3.13)

where

b1(s) = s + λ + μ1 + γ, b2(s) = s + λ + μ2 + γ, c(s) = s + λ + γ

and

|D(s)| = s3 + (3λ+3γ +μ)s2+(3λ2 + λ(μ+μ2 + 6γ )+2μγ + μ1μ2 + 3γ 2)s+ λ3 + (μ2 + 3γ )λ2 + (μ1γ + 2μ2γ + 3γ 2)λ + μ1μ2γ + μγ 2 + γ 3.

The characteristic roots of the matrix B are given by

|D(s)| = 0. (3.14)

Let sk, (k = 1, 2, 3) be the characteristic roots of Eq. (3.13). Such roots are

s1 = −(λ + μ2 + γ ),

s2, s3 =−(2λ + μ1 + 2γ ) ±

√4λμ1 + μ21

2.

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We observe that b∗i j (s) are rational algebraic functions in s. The cofactor of the (i, j)thelement of (s I − B) is a polynomial of degree 2−|i − j |. Since the characteristc rootssk, (k = 1, 2, 3) of B are all real and distinct, the inverse transform bi j (t) of b∗i j (s)can be obtained by partial fraction decomposition. Now using Eq. (3.13), we get

eT (s I − B)−1e1 =3∑j=1

b∗j1(s), (3.15)

eT (s I − B)−1e2 =3∑j=1

b∗j2(s) (3.16)

and

eT (s I − B)−1[μ2e2 + μ1e3] =⎡⎣μ2 3∑

j=1b∗j2(s) + μ1

3∑j=1

b∗j3(s)

⎤⎦ . (3.17)

Using Eqs. (3.15)–(3.17) in Eq. (3.12), we get

P̂23(s) =[s + γ − 1

2

[w −

√w2 − α2 − 2λ

]+ c∗2(s)

]−1

×[1+η

[P̂04(s)+ P̂14(s)+ P̂24(s)

]+η

∞∑i=1

P̂(i+2)4(s)(

w−√w2−α22λ

)i

− c∗0(s) − ηc∗0(s)P̂04(s) − ηc∗1(s)P̂14(s)], (3.18)

where

c∗0(s) = (s + γ )[b∗11(s) + b∗21(s)],c∗1(s) = (s + γ )[b∗12(s) + b∗22(s)]

and

c∗2(s) = (s + γ )[μ2(b∗12(s) + b∗22(s)

)+ μ1(b∗13(s) + b∗23(s) + b∗33(s))] .After some simple algebraic manipulations, Eq. (3.18) reduces to

P̂23(s) = 2w +

√w2 − α2

{1 + η[P̂04(s) + P̂14(s) + P̂24(s)]

+ η∞∑i=1

P̂(i+2)4(s)(

w −√

w2 − α22λ

)i− c∗0(s) − ηc∗0(s)P̂04(s) − ηc∗1(s)P̂14(s)

⎫⎬⎭

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×[1 −(μ

λ

) 12 w −

√w2 − α2α

(1 − c

∗2(s)

μ

)]−1,

P̂23(s) =[2

α

(w −√w2 − α2α

)(1 + η

[P̂04(s) + P̂14(s) + P̂24(s)

]

+ η∞∑i=1

P̂(i+2)4(s)(w −√w2 − α2

2λ

)i − c∗0(s) − ηc∗0(s)P̂04(s) − ηc∗1(s)P̂14(s))]

×[ ∞∑m=0

(μλ

)m2(w −√w2 − α2

α

)m]×⎡⎣ ∞∑m=0

m∑k=0

(−1)k(m

k

)( c∗2(s)μ

)k⎤⎦ .

On Laplace inversion, we get an explicit expression for P23(t) as

P23(t)

=( ∞∑m=0

(μλ

)m2

m∑k=0

(−1)k(m

k

)(1

μ

)k)

×(∫ t

0c∗k2 (t − u) exp{−(λ + μ + γ )u}[Im(αu) − Im+2(αu)]du

+ η∫ t0c∗k2 (t − u)

[∫ u0

P04(u − v) exp{−(λ + μ + γ )v}[Im(αv) − Im+2(αv)]dv]du

+ η∫ t0c∗k2 (t − u)

[∫ u0


+ η∫ t0c∗k2 (t − u)

[∫ u0


+ η∫ t0c∗k2 (t − u)

×[∫ u

0

∞∑i=1

P(i+2)4(u − v)β−i exp{−(λ + μ + γ )v}[Im+i (αv) − Im+i+2(αv)]dv]du

−∫ t0c∗k2 (t − u)

[∫ u0

c0(u − v) exp{−(λ + μ + γ )v}[Im(αv) − Im+2(αv)]dv]du

− η∫ t0c∗k2 (t − u)P04(t − u)

×[∫ u

0c0(u − v) exp{−(λ + μ + γ )v}[Im(αv) − Im+2(αv)]dv

]du

− η∫ t0c∗k2 (t − u)P14(t − u)

×[∫ u

0c1(u − v) exp{−(λ + μ + γ )v}[Im(αv) − Im+2(αv)]dv

]du

), (3.19)

where c∗k2 is the k-fold convolution of c2(t) with itself and c∗0

2 (t) = δ(t).

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3.3 Evaluation of P00(t), P11(t), P12(t)

Using Eq. (3.13) in Eq. (3.8), we have

P̂00(s) =[η P̂04(s) + 1

]b∗11(s) +

[μ2 P̂23(s) + η P̂14(s)

]b∗12(s) + μ1 P̂23(s)b∗13(s),

(3.20)

P̂11(s) =[η P̂04(s) + 1

]b∗21(s) +

[μ2 P̂23(s) + η P̂14(s)

]b∗22(s) + μ1 P̂23(s)b∗23(s)

(3.21)

and

P̂12(s) =[η P̂04(s) + 1

]b∗31(s) +

[μ2 P̂23(s) + η P̂14(s)

]b∗32(s) + μ1 P̂23(s)b∗33(s).

(3.22)

Using Eqs. (3.20)–(3.22) and inverting, we obtain

P00(t) = b11(t)+∫ t0

ηP04(u)b11(t − u)du+∫ t0

(μ2P23(u)+ηP14(u))b12(t − u)du

+∫ t0

μ1P23(u)b13(t − u)du, (3.23)

P11(t) = b21(t)+∫ t0

ηP04(u)b21(t − u)du+∫ t0

(μ2P23(u)+ηP14(u))b22(t − u)du

+∫ t0

μ1P23(u)b23(t − u)du, (3.24)

P12(t) = b31(t)+∫ t0

ηP04(u)b31(t − u)du+∫ t0

(μ2P23(u)+ηP14(u))b32(t − u)du

+∫ t0

μ1P23(u)b33(t − u)du. (3.25)

3.4 Evaluation of Pn4(t), n ≥ 1

The transient-state system size probabilities Pn4(t), n ≥ 1 are obtained using con-tinued fractions. Let f̂s denote the Laplace transform of f (t). On taking Laplacetransform of Eq. (2.7), we have

P̂n4(s)

P̂(n−1)4(s)= λ

(s + λ + η + nξ) − (n + 1)ξ P̂(n+1)4(s)P̂n4(s)

,

= λ(s + λ + η + nξ) − (n+1)ξλ

s+λ+η+(n+1)ξ− (n+2)ξλs+λ+η+(n+2)ξ−...· (3.26)

123


The well known identity of confluent hypergeometric function from Lorentzen andWaadeland (1992) is

1F1(a + 1; c + 1; z)1F1(a; c; z) =

c

c − z+(a + 1)z

c − z + 1+(a + 2)z

c − z + 2+ . . . ,

which can be rewritten as

c1F1(a; c; z)

1F1(a + 1; c + 1; z) − (c − z) =(a + 1)z

c − z + 1+(a + 2)z

c − z + 2+ . . . . (3.27)

Using the identity, (3.26) takes the form

P̂n4(s)

P̂(n−1)4(s)= λ

ξ

1F1(n+1; s+η

ξ+n+1;− λ

ξ

)(s+ηξ

+n)1F1(n;(s+ηξ

+n);− λ

ξ

) ·

Invoking of the above equation, for n = 1, 2, 3, . . . , gives

P̂n4(s) =(

λ

ξ

)n 1∏ni=1(s+ηξ

+ i) 1F1

(n + 1; s+η

ξ+ n + 1;−λ

ξ

)1F1(1;(s+ηξ

+ 1)

;−λξ

) P̂04(s), (3.28)= φ̂n(s)P̂04(s). (3.29)

On Laplace inversion, we obtain

Pn4(t) = φn(t) ∗ P04(t), (3.30)

where φn(t) is the inverse Laplace transform of φ̂n(s) and ∗ denotes convolution.

3.4.1 Evaluation of P04(t)

For any t ≥ 0,

P00(t) + P11(t) + P12(t) +∞∑n=2

Pn3(t) +∞∑n=0

Pn4(t) = 1

and its Laplace transform is of the form

P̂00(s) + P̂11(s) + P̂12(s) +∞∑n=2

P̂n3(s) +∞∑n=0

P̂n4(s) = 1s. (3.31)

Substituting (3.29) in (3.31), we get

P̂00(s) + P̂11(s) + P̂12(s) +∞∑n=2

P̂n3(s) = 1s

− P̂04(s)[1 +

∞∑n=1

φ̂n(s)

]. (3.32)

123


The Laplace transform of (2.6) together with (3.29) for n = 1 and (3.31) yield

P̂04(s) = γs

[s + λ + η + γ − ξ φ̂1(s) + γ

∞∑n=1

φ̂n(s)

]−1. (3.33)

After considerable simplification of (3.33), we get

P̂04(s) = γs

∞∑k=0

(−1)k(s + λ + η + γ )k+1

[ ∞∑i=1

(γ − δ1iξ)φ̂i (s)]k

. (3.34)

On Laplace inversion, we obtain

P04(t)=γ∞∑k=0

(−1)k∫ t0

⎧⎨⎩exp{−(λ+η+γ )y} y

k

k! ∗[ ∞∑i=1

(γ −δ1iξ)φi (y)]∗k⎫⎬⎭ dy,(3.35)

where φn(t) is the inverse Laplace transform of φ̂n(s) and ∗ denotes convolution.

3.4.2 Expression of φn(t)

From Eq. (3.30), we have

φ̂n(s) =(

λ

ξ

)n 1∏ni=1(s+ηξ

+ i) 1F1

(n + 1; s+η

ξ+ n + 1;−λ

ξ

)1F1(1;(s+ηξ

+ 1)

;−λξ

) . (3.36)

The confluent hypergeometric function is defined by the power series as

1F1(n + 1; s+η

ξ+ n + 1;−λ

ξ

)∏n

i=1(s+ηξ

+ i) = ξn ∞∑

k=0

(n+kk

)(−λ)k∏n+k

i=1 (s + η + iξ).

By resolving into partial fractions, we have

1F1(n + 1; s+η

ξ+ n + 1;−λ

ξ

)∏n

i=1(s+ηξ

+ i)

= ξ∞∑k=0

(n + kk

)(−λξ

)k n+k∑i=1

(−1)i−1(i − 1)!(n + k − i)!

1

s + η + iξ . (3.37)

123


Also,

1F1

(1;(s + η

ξ+ 1)

;−λξ

)=

∞∑k=0

(−λ)k∏n+ki=1 (s + η + iξ)

=∞∑k=0

(−λ)k âk(s), â0(s) = 1,

where

âk(s) = 1∏ki=1(s + η + iξ)

= 1ξ k−1

k∑r=1

(−1)r−1(r − 1)!(k − r)!

1

s + η + rξ , k = 1, 2, 3, . . . .

Using the identity given in Gradshteyn and Ryzhik (2007),

[1F1

(1;(s + η

ξ+ 1)

;−λξ

)]−1=

∞∑k=0

b̂k(s)λk, (3.38)

where b̂0(s) = 1 and for k = 1, 2, 3, . . . ,

b̂k(s) =

∣∣∣∣∣∣∣∣∣∣∣∣∣

â1(s) 1 . . .â2(s) â1(s) 1 . . .â3(s) â2(s) â1(s) . . .

......

...

âk−1(s) âk−2(s) âk−3(s) . . . â1(s) 1âk(s) âk−1(s) âk−2(s) . . . â2(s) â1(s)

∣∣∣∣∣∣∣∣∣∣∣∣∣=

k∑i=1

(−1)i−1âi (s)b̂k−i (s).

By substituting (3.37) and (3.38) in (3.36), we obtain,

φ̂n(s) = λn∞∑j=0

(−λ) j(n + j

j

)ân+ j (s)

∞∑k=1

λk b̂k(s).

On Laplace inversion,

φn(t) = λn∞∑j=0

(−λ) j(n + j

j

)an+ j (t) ∗

∞∑k=1

λkbk(t), (3.39)

123


where

ak(t) = 1ξ k−1

k∑r=1

(−1)r−1(r − 1)!(k − r)! exp{−(η + rξ)t}, k = 1, 2, 3, . . . ,

bk(t) =k∑

i=1(−1)i−1ai (t) ∗ bk−i (t), k = 2, 3, 4, . . . ; b1(t) = a1(t).

Thus Eqs. (3.6), (3.19), (3.23)–(3.25), (3.30), (3.35) and (3.39) completely determineall the time-dependent system size probabilities.

3.4.3 Special cases

Case 3.1 For γ = η = 0,Equation (3.6) reduces to

P(n+2)3(t) = nβn∫ t0exp{−(λ + μ)(t − y)} In(α(t − y))

(t − y) P23(y)dyfor n = 1, 2, 3, . . . , (3.40)

and Eq. (3.19) reduces to

P23(t) =∞∑

m=0

(μλ

)m2

m∑k=0

(−1)k(m

k

)(1

μ

)k ∫ t0c∗k2 (t − u)

×[exp{−(λ + μ)u}[Im(αu) − Im+2(αu)]

−∫ u0

c0(u − v) exp{−(λ + μ)v}[Im(αv) − Im+2(αv)]dv]du. (3.41)

Also Eqs. (3.23), (3.24) and (3.25) reduces to

P00(t) = b11(t) +∫ t0

[μ2b12(t − u) + μ1b13(t − u)]P23(u)du, (3.42)

P11(t) = b23(t) +∫ t0

[μ2b22(t − u) + μ1b23(t − u)]P23(u)du, (3.43)

P12(t) = b31(t) +∫ t0

[μ2b32(t − u) + μ1b33(t − u)]P23(u)du. (3.44)

If γ = η = 0, the results of ourmodel resembleswith themodel studied byDharmaraja(2000). Equations (3.40), (3.41), (3.42), (3.43) and (3.44) coincides with Eqs. (3.10),(3.15), (3.16), (3.17) and (3.18), respectively in Dharmaraja (2000).

123


Case 3.2 We observe that our model gets reduced to the one studied by Ammar(2014a).

For λp = λ and α = 0, Eqs. (2.12), (2.24), (2.25), (2.26) and (2.27) in Ammar(2014a) coincides with Eqs. (3.40), (3.41), (3.42), (3.43) and (3.44), respectively.

Case 3.3 Also if there is no catastrophe, i.e., (ξ = 0) and using the special casec = 2 in the model analyzed by Dharmaraja and Kumar (2015), then for γ = η = 0,our model reduces to the model in Dharmaraja and Kumar (2015). Equations (3.40),(3.41), (3.42), (3.43) coincides with Eqs. (16), (32), (33) (k = 0, 1), respectively inDharmaraja and Kumar (2015).

Case 3.4 If there is no catastrophe, i.e., (γ = 0) and the arrivals do not opt the fasterand slower servers probabilistically as described in Kumar et al. (2007), the transient-state probabilities of our model agree with that of Kumar et al. (2007). For λp = λ andα = 0, Eqs. (2.12), (2.29), (2.30), (2.31) and (2.32) in Kumar et al. (2007) coincideswith Eqs. (3.40), (3.41), (3.42), (3.43) and (3.44), respectively.

4 Moments

In this section, we obtain the time-dependent first order and second order moments ofthe system.

4.1 Mean

Let X (t) denote the number of customers in the system at time t . The average numberof customers in the system at time t is given by

m(t) = E(X (t)) = P11(t) + P12(t) +∞∑n=0

(n + 2)P(n+2)3(t) +∞∑n=1

nPn4(t), (4.1)

and

m′(t) = P ′11(t) + P

′12(t) +

∞∑n=0

(n + 2)P ′(n+2)3(t) +∞∑n=1

nP′n4(t).

From Eqs. (2.2)–(2.5) and (2.7), after considerable mathematical manipulations, theabove equation will lead to the following differential equation

m′(t) = −γm(t) + (λ − μ) + μP00(t) + μ1P11(t) + μ2P12(t) + (γ − ξ)

×∞∑n=1

nPn4(t) + μ∞∑n=0

Pn4(t).

123


Therefore,

m(t) = 1γ

(λ − μ)(1 − e−γ t ) + μ∫ t0

P00(u)e−γ (t−u)du

+μ1∫ t0


+μ2∫ t0

P11(u)e−γ (t−u)du + (γ − ξ)

∞∑n=1

n∫ t0

Pn4(u)e−γ (t−u)du

+μ∞∑n=0

∫ t0

Pn4(u)e−γ (t−u)du, (4.2)

where P00(t), P12(t), P11(t) and Pn4(t) are given in Eqs. (3.23)–(3.25) and (3.30),respectively.

4.2 Variance

The variance number of customers in the system at time t is given by

Var(X (t)) = E(X2(t)) − [E(X (t))]2,Var(X (t)) = h(t) − [m(t)]2, (4.3)

where

h(t) = E(X2(t)) = P12(t) + P11(t) +∞∑n=0

(n + 2)2P(n+2)3(t) +∞∑n=1

n2Pn4(t).

Then

h′(t) = P ′12(t) + P

′11(t) +

∞∑n=0

(n + 2)2P ′(n+2)3(t) +∞∑n=1

n2P′n4(t).

From Eqs. (2.2)–(2.5) and (2.7), after considerable mathematical manipulations, theabove equation will lead to the following differential equation

h′(t) = −γ h(t) + 2(λ − μ)m(t) + (λ + μ) − μP00(t) − μP04(t) + μ1P12(t)

+μ2P11(t) + γ∞∑n=1

n2Pn4(t) +∞∑n=1

(2n − 1)(μ − nξ)Pn4(t). (4.4)

123


Therefore,

h(t) = 1γ

(λ + μ)(1 − e−γ t ) + 2(λ − μ)∫ t0m(u)e−γ (t−u)du

−μ∫ t0

[P00(u) + P04(u)]e−γ (t−u)du

+μ1∫ t0

P12(u)e−γ (t−u)du + μ2

∫ t0


+ γ∞∑n=1

n2∫ t0


+∞∑n=1

(2n − 1)(μ − nξ)∫ t0

Pn4(u)e−γ (t−u)du.

Substituting the above equation in (4.3), we get

Var(X (t)) = 1γ

(λ + μ)(1 − e−γ t ) + 2(λ − μ)∫ t0m(u)e−γ (t−u)du

−μ∫ t0

[P00(u) + P04(u)]e−γ (t−u)du

+μ1∫ t0

P12(u)e−γ (t−u)du + μ2

∫ t0


+ γ∞∑n=1

n2∫ t0


+∞∑n=1

(2n − 1)(μ − nξ)∫ t0

Pn4(u)e−γ (t−u)du − [m(t)]2,

where m(t), P00(t), P12(t), P11(t), P04(t) and Pn4(t) are given in Eqs. (4.2), (3.23)–(3.25), (3.35) and (3.30) respectively.

5 Steady-state probabilities

In this section, we shall discuss the behaviour of the steady-state probabilities of thetwo-heterogeneous servers queue with system disaster, server repair and impatientcustomers. Let

Pnj = limt→∞Prob{X (t) = n,Y (t) = j} (n, j) ∈ S.

123


From Eq. (3.18), for γ, η > 0, we get

P̂23(s) =[s + γ − 1

2

[w −

√w2 − α2 − 2λ

]+ c∗2(s)

]−1

×[1+η

[P̂04(s)+ P̂14(s)+ P̂24(s)

]+η

∞∑i=1

P̂(i+2)4(s)(

w−√w2−α22λ

)i

− c∗0(s) − ηc∗0(s)P̂04(s) − ηc∗1(s)P̂14(s)].

By using Tauberian theorem, we get

P23 =[[

λ3 + (μ2 + 3γ )λ2 + (μ1γ + 2μ2γ + 3γ 2)λ + μ1μ2γ + μγ 2 + γ 3]

×[γ − 1

2((λ + μ + γ ) −

√(λ + μ + γ )2 − α2 − 2λ)

]

+ γ[μ1μ2(λ + μ2 + γ ) + (λ + γ )(λ + μ2 + γ )μ2 + μ1μ2(λ + μ1 + γ )

+ λμ1μ2 + μ1(λ + γ )(λ + μ1 + γ ) − λμ21]]−1

×[[

ηP04+ηP14+ηP24+η∞∑i=1

P(i+2)4( (λ+μ+γ )−√(λ+μ+γ )2−α2

2λ

)i]

×[λ3 + (μ2 + 3γ )λ2 + (μ1γ + 2μ2γ + 3γ 2)λ + μ1μ2γ + μγ 2 + γ 3]−[ηγ (λ+μ2+γ )(2λ+μ1+γ )]P04 − [ηγ (λ+μ2+γ )(λ+μ1+γ )]P14

].

(5.1)

Taking Laplace transform of (3.6) and after considerable simplifications, we get forn ≥ 1

P̂(n+2)3(s)

=[√

(s + λ + μ + γ )2 − α2[(s + λ + μ + γ ) +

√(s + λ + μ + γ )2 − α2

]n]−1

×[[

(2λ)n√

(s + λ + μ + γ )2 − α2]P̂23(s)

+[η

∞∑i=1

(2λ)n−i((s + λ + μ + γ ) +

√(s + λ + μ + γ )2 − α2

)i

×(1 − α2i

((s + λ + μ + γ ) +

√(s + λ + μ + γ )2 − α2

)−2i)]P̂(i+2)4(s)

].

123


Then, again by using Tauberian theorem, we obtain

P(n+2)3 =[√

(λ + μ + γ )2 − α2[(λ + μ + γ ) +

√(λ + μ + γ )2 − α2

]n]−1

×[[

(2λ)n√

(λ + μ + γ )2 − α2]P23

+[η

∞∑i=1

(2λ)n−i((λ + μ + γ ) +

√(λ + μ + γ )2 − α2

)i

×(1 − α2i

((λ + μ + γ ) +

√(λ + μ + γ )2 − α2

)−2i)]P(i+2)4

]. (5.2)

In a similar way, (3.20) gives

P00 =[λ3 + (μ2 + 3γ )λ2 + (μ1γ + 2μ2γ + 3γ 2)λ + μ1μ2γ + μγ 2 + γ 3

]−1

×[[η(λ + μ2 + γ )(λ + μ1 + γ )]P04 + [μ1μ2(λ + μ2 + γ )]P23

+[μ1η(λ + μ2 + γ )]P14 + [μ1μ2(λ + μ1 + γ )]P23]. (5.3)

From (3.21), we have

P11 =[λ3 + (μ2 + 3γ )λ2 + (μ1γ + 2μ2γ + 3γ 2)λ + μ1μ2γ + μγ 2 + γ 3

]−1

×[[λη(λ + μ2 + γ )]P04 + [μ2(λ + γ )(λ + μ2 + γ )]P23

+ [η(λ + γ )(λ + μ2 + γ )] P14 + [λμ1μ2]P23]. (5.4)

Also from (3.22), we yield

P12 = [μ1(λ + γ )(λ + μ1 + γ ) − λμ21]P23

λ3 + (μ2 + 3γ )λ2 + (μ1γ + 2μ2γ + 3γ 2)λ + μ1μ2γ + μγ 2 + γ 3 . (5.5)

Using (3.28), we have for n = 1, 2, 3, . . . ,

Pn4 =[(λ

ξ

)n 1∏ni=1(

ηξ

+ i)1F1(n + 1; η

ξ+ n + 1;−λ

ξ

)1F1(1;(

ηξ

+ 1);−λ

ξ

) ]P04. (5.6)

123


From (3.34), we get

P04 = γ∞∑k=0

(−1)k(λ + η + γ )k+1

⎡⎣ ∞∑i=1

(γ − δ1i ξ)(λ

ξ

)n 1∏ni=1(

ηξ

+ i) 1F1

(n + 1; η

ξ+ n + 1;− λ

ξ

)1F1(1;(

ηξ

+ 1); − λ

ξ

)⎤⎦k

.

(5.7)

6 Performance measures

In this section, we obtain some important performance measures of the system.

6.1 Probability of arriving customers joining the queue

The probability that an arriving customer is required to join the queue at time t is givenby

P(X (t) ≥ 2)= P23(t) +

∞∑n=1

nβn∫ t0exp{−(λ + μ + γ )(t − y)} In(.)

(t − y) P23(y)dy

+ η∫ t0exp{−(λ + μ + γ )(t − y)}

∞∑i=1

βn−i P(i+2)4(y)[In−i (.) − I(n+i)(.)]dy.

(6.1)

Similarly, the steady-state probability that an arriving customer joins the queue is

limt→∞ P(X (t) ≥ 2) =

∞∑n=0

P(n+2)3,

= P23+∞∑n=1

[√(λ+μ+γ )2 − α2 ×

[(λ+μ+γ )+

√(λ+μ+γ )2 − α2

]n]−1

×[[

(2λ)n√

(λ + μ + γ )2 − α2]P23

+[η

∞∑i=1

(2λ)n−i((λ + μ + γ ) +

√(λ + μ + γ )2 − α2

)i

×(1 − α2i

((λ + μ + γ ) +

√(λ + μ + γ )2 − α2

)−2i)]P(i+2)4

]. (6.2)

123


6.2 The number of busy servers

Let B(t) denote the number of busy servers at time t . The probability that the systemhas n busy servers is given as

P(B(t) = n) ={P(X (t) = 1) = P11(t) + P12(t), for n = 1P(X (t) > 1) =∑∞n=0 P(n+2)3(t), for n = 2 .

For γ, η > 0, the corresponding steady-state probabilities are obtained as

limt→∞ P(B(t) = 1)

=[λ3 + (μ2 + 3γ )λ2 + (μ1γ + 2μ2γ + 3γ 2)λ + μ1μ2γ + μγ 2 + γ 3

]−1

×[[λη(λ + μ2 + γ )]P04 + [η(λ + γ )(λ + μ2 + γ )]P14

+[μ1(λ + γ )(λ + μ1 + γ ) − λμ21 + λμ1μ2 + μ2(λ + γ )(λ + μ2 + γ )]P23](6.3)

and

limt→∞ P(B(t) = 2)

= P23 +∞∑n=1

[√(λ + μ + γ )2 − α2

×[(λ + μ + γ ) +

√(λ + μ + γ )2 − α2

]n]−1

×[[

(2λ)n√

(λ + μ + γ )2 − α2]P23

+[η

∞∑i=1

(2λ)n−i((λ + μ + γ ) +

√(λ + μ + γ )2 − α2

)i

×(1 − α2i

((λ + μ + γ ) +

√(λ + μ + γ )2 − α2

)−2i)]P(i+2)4

]. (6.4)

6.3 The mean number of busy servers

Furthermore, the mean number of busy servers at time t is given by

E(B(t)) = P11(t) + P12(t) + 2∞∑n=0

P(n+2)3(t).

123


This can be simplified as

E(B(t)) = 2[1 − P00(t) −

∞∑n=0

Pn4(t)

]− [P11(t) + P12(t)].

For γ, η > 0, the corresponding steady-state result is given as

limt→∞ E(B(t))

= 2(1 −[λ3 + (μ2 + 3γ )λ2 + (μ1γ + 2μ2γ + 3γ 2)λ + μ1μ2γ + μγ 2 + γ 3

]−1

×[[η(λ + μ2 + γ )(λ + μ1 + γ )]P04 + [μ1μ2(λ + μ2 + γ )]P23

+ [μ1η(λ + μ2 + γ )]P14 + [μ1μ2(λ + μ1 + γ )]P23]

−[(λ

ξ

)n 1∏ni=1(

ηξ

+ i)1F1(n + 1; η

ξ+ n + 1;−λ

ξ

)1F1(1;(

ηξ

+ 1);−λ

ξ

) ]P04− γ

∞∑k=0

(−1)k(λ + η + γ )k+1

×⎡⎣ ∞∑

i=1(γ − δ1iξ)

(λξ

)n 1∏ni=1(

ηξ

+ i)1F1(n + 1; η

ξ+ n + 1;−λ

ξ

)1F1(1;(

ηξ

+ 1);−λ

ξ

)⎤⎦k )

−[λ3 + (μ2 + 3γ )λ2 + (μ1γ + 2μ2γ + 3γ 2)λ + μ1μ2γ + μγ 2 + γ 3

]−1

×[[λη(λ + μ2 + γ )]P04 + [η(λ + γ )(λ + μ2 + γ )]P14

+ [μ1(λ + γ )(λ + μ1 + γ ) − λμ21 + λμ1μ2+ μ2(λ + γ )(λ + μ2 + γ )]P23

]. (6.5)

6.4 Probability that the system is empty

The probability that the system is empty at time t is given by

= P00(t) + P04(t),= b11(t) +

∫ t0

ηP04(u)b11(t − u)du +∫ t0

(μ2P23(u) + ηP14(u))b12(t − u)du

123


+∫ t0

μ1P23(u)b13(t − u)du + γ∞∑k=0

(−1)k∫ t0

×⎧⎨⎩exp{−(λ + η + γ )y} y

k

k! ∗[ ∞∑i=1

(γ − δ1iξ)φi (y)]∗k⎫⎬⎭ dy (6.6)

using Eqs. (3.23) and (3.35).

6.5 Probability that the system is in breakdown

The probability that the system is in breakdown at time t is given by

=∞∑n=0

Pn4(t),

=⎡⎣γ ∞∑

k=0(−1)k

∫ t0

⎧⎨⎩exp{−(λ + η + γ )y} y

k

k! ∗[ ∞∑i=1

(γ − δ1iξ)φi (y)]∗k⎫⎬⎭ dy

⎤⎦

×(1 + φn(t)

), (6.7)

where φn(t) are given in Eq. (3.39).

7 Conclusion and future work

In this paper, we have analyzed the M/M/2 queueing model with two-heterogeneousservers subject to system disaster, server repair and customers’ impatience wherecustomers become impatient when the system is down.

The explicit expressions are obtained for the time-dependent probabilities of theunderlying queueing model. It would be interesting to investigate a similar model withc-heterogeneous servers, system disaster, server repair and customers’ impatience.Further, one can evaluate the optimality of service rates and repair rate to minimizethe waiting time of the customers in the system.

Acknowledgements The authors wish to thank the anonymous referees for their careful review and valu-able suggestions that led to considerable improvement in the presentation of this paper.

References

Ammar SI (2015) Transient analysis of an M/M/1 queue with impatient behavior and multiple vacations.Appl Math Comput 260:97–105

Ammar SI (2014a) Transient analysis of a two-heterogeneous servers queue with impatient behavior. JEgypt Math Soc 22(1):90–95

Ammar SI (2014b) Transient behavior of a two-processor heterogeneous system with catastrophes, serverfailures and repairs. Appl Math Model 38(7–8):2224–2234

123


Armony M, Ward AR (2010) Fair dynamic routing in large-scale heterogeneous-server systems. Oper Res58(3):624–637 MR2680568

Chen A, Renshaw E (1997) The M/M/1 queue with mass exodus and mass arrivals when empty. J ApplProbab 34(1):192–207

Dharmaraja S (2000) Transient solution of a two-processor heterogeneous system. Math Comput Model32(10):1117–1123

Dharmaraja S, Kumar R (2015) Transient solution of a Markovian queuing model with heterogeneousservers and catastrophes. Opsearch (in press)

Gelenbe E (1991) Product-form queueing networks with negative and positive customers. J Appl Probab28(3):656–663

Gradshteyn IS, Ryzhik IM (2007) Table of integrals, series, and products, 7th edn. Elsevier/Academic Press,Amsterdam

Ibrahima R, LEcuyerb P, Shenc H (2016) Inter-dependent, heterogeneous, and time-varying service-timedistributions in call centers. Eur J Oper Res 250(2):480–492 (to appear in)

Ke JC, Liu TH, Wu CH (2015) An optimum approach of profit analysis on the machine repair system withheterogeneous repairmen. Appl Math Comput 253:40–51 MR3312697

Kumar BK et al (2007) Transient analysis of a single server queue with catastrophes, failures and repairs.Queueing Syst 56(3–4):133–141

Kumar BK, Madheswari SP, Venkatakrishnan KS (2007) Transient solution of an M/M/2 queue withheterogeneous servers subject to catastrophes. Int J Inform Manag Sci 18(1):63–80

Krishna Kumar B, Vijayakumar A, Sophia S (2008) Transient analysis for state-dependent queues withcatastrophes. Stoch Anal Appl 26(6):1201–1217

Lippolt CR, Arnold D, Dörrsam V (2003) Analysis of a single stage production system with heterogeneousmachines. OR Spectrum 25(1):97–107 MR1957859 (2003m:90030)

Lorentzen L, Waadeland H (1992) Continued fractions with applications. Studies in computational mathe-matics, vol 3. North-Holland, Amsterdam

Sudhesh R, Sebasthi Priya R, Lenin RB (2016) Analysis of N-policy queues with disastrous breakdown.TOP (accepted: 8 February)

Sudhesh R (2010) Transient analysis of a queue with system disasters and customer impatience. QueueingSyst 66(1):95–105

Trivedi KS (2002) Probability and statistics with reliability, queuing, and computer science applications,2nd edn. Wiley, New York

Yechiali U (2007) Queues with system disasters and impatient customers when system is down. QueueingSyst 56(3–4):195–202

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Transient analysis of a two-heterogeneous servers queue with system disaster, server repair and customers' impatienceAbstract1 Introduction2 Model description2.1 The service discipline2.2 Governing equations

3 Transient behaviour3.1 Evaluation of P(n+2)3(t),n13.2 Evaluation of P23(t)3.3 Evaluation of P00(t),P11(t),P12(t)3.4 Evaluation of Pn4(t), n13.4.1 Evaluation of P04(t)3.4.2 Expression of φn(t)3.4.3 Special cases

4 Moments4.1 Mean4.2 Variance

5 Steady-state probabilities6 Performance measures6.1 Probability of arriving customers joining the queue6.2 The number of busy servers6.3 The mean number of busy servers6.4 Probability that the system is empty6.5 Probability that the system is in breakdown

7 Conclusion and future workAcknowledgementsReferences

transient analysis of a two-heterogeneous servers queue with...

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