transientanalysis of m x m x g 1 retrial queueing system ... · li et al. (2017) and zaiming liu et...

International Journal of Applied Engineering Research, ISSN 0973-4562 Volume 13, Number 11 (2018) pp. 8783–8795© Research India Publications, http://www.ripublication.com

Transient Analysis of M [X1], M [X2]/G1, G2/1 retrial queueing system withpriority services, Collisions, Orbital Search, Working Breakdown, Startup/Close down time, Feedback, modified Bernoulli vacation and Balking

G. AyyappanDepartment of Mathematics, Pondicherry Engineering College, India.

J. UdayageethaDepartment of Mathematics, Perunthalaivar Kamarajar Arts College, India.

Abstract: This paper considers M [X1], M [X2]/G1, G2/1general retrial queueing system with priority services. Theserver serves two types of customers and follows the pre-emptive priority rule subject to collisions, orbital search,working breakdown, startup/closedown times and modi-fied Bernoulli vacation with general (arbitrary) vacationperiods. After completing the service, if there are no prior-ity customers present in the system then the server may gofor a vacation or close down the system. On completion ofthe close down, the server needs some time to set up thesystem. The priority customers who find the server busyare queued in the system. If a low-priority customer findsthe server busy, they are routed to a retrial (orbit) queue thatattempts to get the service. The system may become defec-tive at any point of time when it is in operation. However,when the system is defective, instead of stopping servicecompletely, the service continues only to the high prioritycustomers at a slower rate.

We consider balking to occur at the low priority customerwhile the server is busy or idle. If, the server is busy withthe low priority customer an arriving low priority customermay collide with the customer present in the system andboth will be shifted to orbit. Using the supplementary vari-able technique, the steady-state distributions of the serverstate and the number of customers in the orbit are obtained.The stochastic decomposition property is investigated. Fur-ther, some particular cases of interest are discussed. Finally,numerical illustrations are provided.

AMS subject classification: 60K25, 68M30.Keywords: Collisions, Orbital Search, Priority QueueingSystems, Retrial, Bernoulli Vacations, Working Break-down, Startup/close down time, Balking.

INTRODUCTION

Pre-emptive priority queue was integrated by Jaiswal (1968).Several authors studied about priority queueing systemwith retrial, where the customers are joined the orbit andretry for their service. The perfectly reliable servers are notpossible in real world phenomena. In fact, the servers maywell be subject to lengthy and unpredictable breakdownswhile serving a customer. For example, in manufacturingsystems the machine may breakdown due to machine orjob related problems. This results in a period of unavail-able time until the servers are repaired. Such a system withrepairable server has been studied as a queueing modeland reliability model by many authors. However, they haveassumed that the server breaksdown even when the serveris idle. In our paper, we have assumed that the server failsonly in the operational state.

Studying queueing models with server breakdown, itis generally assumed that the server stops service whenthe server breaks down. However, in most of the modelsconsidered so far of queueing systems with server break-downs, the underlying assumption has been that a server


breakdown disrupts the service completely in the system.In computer, the presence of a virus in the system mayslow down the performance of the computer system. Thecomputer system may still be able to perform various tasksbut at a considerably slower rate. Here the failure of thecomputer system does not stop the work completely. Moti-vated by this factor, we have therefore considered in thispaper a new class of queueing systems with the workingbreakdown policy with various parameters.

Kalidass et al. (2012) introduced the working break-down policy, in which the server works at a lower servicerate rather than stopping service during the breakdownperiod. In this policy the service can decrease complaintsfrom the customers who should wait for the server tobe, repaired and reduces the cost of waiting customers.Therefore, the working breakdown service is a more rea-sonable repair policy for unreliable queueing systems. TaoLi et al. (2017) and Zaiming Liu et al. (2014) describesmore about working breakdowns. Recently, Kim B.K.et al. (2016) studied the M/G/1 queueing system withdisasters and working breakdowns. They presented cycleanalysis, and derived the system size distribution and thesojourn time distribution. Cheng-Dar Liou (2013) appliedthe matrix-geometric method to examine an infinite capac-ity Markovian queue with an unreliable server subject toworking breakdowns and impatient customers.

Reflecting more practical situations, the server mayturns off his service by a close down time and a startup time is required before starting for service. Dim-itriou et al. (2009) studied about the queueing model withstartup/closedown time and retrial customers. Customerimpatience can be viewed as a potential loss of customers.Recenty, Yang et al. (2017) studied the analysis of a finitecapacity system with working breakdown and retention ofimpatient customers.

In physics a collision occurs when two or more objectshit each other. When objects collide, each object feels aforce for a short amount of time. This force imparts animpulse, or changes the momentum of each of the collidingobjects. A traffic collision occurs when a vehicle collideswith another vehicle, pedestrian, animal, road debris, orother stationary obstruction, such as a tree, pole or building.Traffic collisions often result in injury, death, and prop-erty damage. Most chemical reactions occur only whenreactants collide with one another.

In many situations involving data transmission from var-ious sources there can be problem for a limited numberof channels or other facilities. Uncoordinated attempts by

several sources to use a single server facility can resultin Collision leading to the loss of the transmission andhence the need for retransmission. An important prob-lem concerns the development of workable procedures forremoving the problem and corresponding message delay.Thus retrial queues with collisions are also appropriate formodeling the processes in computer and communicationsnetworks. Krishnakumar et al. (2010) and Peng Y. et al.(2013) studied about collisions with various parameters.Recently, Ayyappan et al. (2017) discussed the priorityretrial queueing system with collision and orbital search.

This paper considers M [X1], M [X2]/G1, G2/1 generalretrial queueing system with priority services. Two dif-ferent sorts of customers arrive at the system in twoindependent compound Poisson processes. Under the pre-emptive priority rule, the server providing general serviceto the arriving customers is subject to collisions, orbitalsearch and modified Bernoulli vacation with general vaca-tion time. We propose a retrial queueing model with theadditional characteristics of server’s working breakdown,repair, startup/closedown times, balking. Arriving high pri-ority customer who find the server busy with high(low)priority customer are queued (pre-empts the low priorityservice) and then are served in accordance with FCFS dis-cipline. The arriving low-priority customer on finding theserver busy cannot be queued they leave the service area andjoin the orbit as retrial customer. After completing service,if there is no high priority customer present in the system,the server may go for a vacation or close down the system.After completing the close down time the server need sometime to set up the system.After completing vacation, repair,setup, if there is no high priority customer present in thesystem the server may search for the low priority customersin the orbit with probability r or remains idle in the system.We consider balking to occur at the low priority customersduring server’s busy or idle period.

The summary of the paper is as follows. Section 1 isan introduction to priority retrial queueing discipline andcomprises literature review. Section 2 deals with modeldescription, notations used, mathematical formulation andgoverning equations of the model. Section 3 elucidatesthe steady state solutions of the system. In section 4,stochastic decomposition is derived. Section 5 demon-strates the performance measures of the model. In Section6, the numerical results and graphs are computed followingwhich the conclusion is given.

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MODEL DESCRIPTION

We consider an unreliable single server retrial queueingmodel with two types of customers namely, high prior-ity and low-priority customers. The basic operation of themodel can be described as:

Arrival and retrial process: Two class of customers arriveat the system in two independent compound Poisson pro-cesses with arrival rate λ1 and λ2 respectively. Let λ1c1,i dt

and λ2c2,i dt (i = 1, 2, 3, . . . ) be the first order proba-bility that a batch of ’i’ customers arrives at the systemduring a short interval of time (t , t + dt), where for 0 ≤c1,i ≤ 1,

∞∑i=1

c1,i = 1, 0 ≤ c2,i ≤ 1,∞∑i=1

c2,i = 1 and

λ1 > 0, λ2 > 0. The arriving high priority customerwho find the server busy is queued and then is served.The arriving low-priority customer on finding the serverbusy, are routed to a retrial queue and they follow con-stant retrial policy that attempts to get the service. Theretrial time is generally distributed with distribution func-tion I (s) and the density function i(s). Let η(x)dx be theconditional probability of completion of retrial during theinterval (x, x + dx] where x is the elapsed retrial time. Ifthe server is busy with low priority customer then the arriv-ing customer may collide with it and both are shifted to theorbit with probability p1.

Service process: If a high priority customer arrives in batchand finds a low priority customer in service, they pre-emptthe low priority customer who is undergoing service; thusthe service of the pre-empted low priority customer beginsonly after the completion of service of all high priority cus-tomers present in the system. The service times for the highpriority and low priority customers are generally(arbitrary)distributed with distribution functions Bi(s) and the den-sity functions bi(s), i = 1, 2 respectively. Let µi(x)dx bethe conditional probability of completion of the high prior-ity and low priority customers service during the interval(x, x + dx], where x is the elapsed service time.

Modified Bernoulli Vacation: After completing all highpriority customers and every service completion of lowpriority customer the server may take a vacation with prob-ability θ or continue the service to the next customer withprobability 1−θ .Vacation time is generally distributed withdistribution function V (s) and the density function v(s).Let β(x)dx be the conditional probability of completion

of vacation during the interval (x, x + dx] where x is theelapsed vacation time.

Working Breakdown state: The server may become inac-tive during busy period. At the time of breakdown the highpriority customer who is in service will get service contin-uosly by lower service rate µ3 and it follows exponentialdistribution. But, at the time of breakdown the low prioritycustomer who is in service will send to the orbit.

Repair Process: The broken down server is sent for repairimmediately so as to regain its functionality with expo-nential repair rate γ . Immediately after returning from therepair, the server starts to serve high priority/low prioritycustomers.

Close down time: After completing vacation, repair, ser-vice, if there are no high priority customers present in thesystem the server will close down the system. Close downtime is generally distributed with distribution function C(s)and the density function c(s). Let φ(x)dx be the conditionalprobability of a completion of close down time during theinterval (x, x+dx] where x is the elapsed close down time.

Startup time: On completion of close down time, theserver takes some time to start up the system to increasethe efficiency of the service. Startup time is generally dis-tributed with distribution function M(s) and the densityfunction m(s). Let δ(x)dx be the conditional probability ofa completion of startup time during the interval (x, x+dx],where x is the elapsed startup time.

Orbital Search: After completing vacation, setup, repairif there is no high priority customer present in the systemthe server may search the orbit to serve the low prioritycustomer with probability r or he may ideally present inthe system.

Balking: If the server is busy or unavailable in the system,an arriving low-priority customers either join the orbit withprobability b or balks with probability (1 − b).

Idle State: After completing the startup or vacation orrepair if there is any high priority customer waiting in thesystem the server starts doing the service. Otherwise theserver is simply present in the system for the customers toarrive.

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FIGURE 1. Flowchart Of Model Description

Definitions and notations

LetN1(t),N2(t) be the queue size and orbit size respectivelyat time t, B0

1 (t),B02 (t) and I 0(t) be the elapsed service time

of the high, low priority customer and retrial time of thelow priority customers, respectively at time t. In addition,let V 0(t), C0(t) and M0(t), be the elapsed vacation time,close down time and setup time of the server respectivelyat time t. The state of the server at time t is given by,

Y (t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0, if the server is in idle state;1, if the server is busy with high priority customer;2, if the server is busy with low priority customer;3, if the server is in vacation;4, if the server is in working breakown state;5, if the server is in repair process;6, if the server is in closedown state;7, if the server is in startup state.

We have I (x), Bi(x), V (x), C(x) and M(x) is continuous

at x = 0, and η(x) = dI (x)

1 − I (x), µi(x)dx = dBi(x)

1 − Bi(x),

β(x)dx = dV (x)

1 − V (x), φ(x)dx = dC(x)

1 − C(x), and δ(x)dx =

dM(x)

1 − M(x), are the first order differential (hazard rate)

functions of I(.), Bi(.), V (.), C(.) and M(.), respectively.

Queue size distribution

Since service time, vacation time, closedown timeand startup time are not exponential, the process{Y (t), N1(t), N2(t)} is not Marovian. In such case weintroduce supplementary variables corresponing to elapsedtimes to make it Markovian (Cox(1955)). Joint distri-butions of the server state and queue size are defined

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as,

I 0,n(s, t) = Pr{Y (t) = 0, N1(t) = 0,

N2(t) = n}, n ≥ 0

P(1)m,n(x, s, t)dx = Pr{Y (t) = 1, x < B0

1 (t) ≤ x + dx,

N1(t) = m, N2(t) = n}, m ≥ 0, n ≥ 0

P(2)0,n(x, s, t)dx = Pr{Y (t) = 2, x < B0

2 (t) ≤ x + dx,

N1(t) = 0, N2(t) = n}, n ≥ 0

V m,n(x, s, t)dx = Pr{Y (t) = 3, x < V 0(t) ≤ x + dx,

N1(t) = m, N2(t) = n}, m ≥ 0, n ≥ 0

Cm,n(x, s, t)dx = Pr{Y (t) = 6, x < C0(t) ≤ x + dx,

N1(t) = m, N2(t) = n}, m ≥ 0, n ≥ 0

Mm,n(x, s, t)dx = Pr{Y (t) = 7, x < M0(t) ≤ x + dx,

N1(t) = m, N2(t) = n}), m ≥ 0, n ≥ 0

Equations Governing the System

The Kolmogorov forward equations which governs themodel:

∂

∂tP (1)

m,n(x, t) + ∂

∂xP (1)

m,n(x, t)

= −(λ1 + λ2 + α + µ1(x))P (1)m,n(x, t)

+ (1 − δm0)λ1

m∑i=1

c1,iP(1)m−i,n(x, t)

+ (1 − δ0n)λ2b

n∑i=1

c2,iP(1)m,n−i(x, t)

+ λ2(1 − b)P (1)m,n(x, t)

+ γQm,n(t); m ≥ 0, n ≥ 0, (1)

∂

∂tVm,n(x, t) + ∂

∂xVm,n(x, t)

= −(λ1 + λ2 + ξ + β(x))Vm,n(x, t)

+ (1 − δm0)λ1

m∑i=1

c1,iVm−i,n(x, t)

+ (1 − δ0n)λ2b

n∑i=1

c2,iVm,n−i(x, t)

+ λ2(1 − b)Vm,n(x, t) + ξVm+1,n(x, t)m ≥ 0,

n ≥ 0, (2)

∂

∂tP

(2)0,n (x, t) + ∂

∂xP

(2)0,n (x, t)

= −(λ1 + λ2 + α + µ2(x))P (2)0,n (x, t)

+ (1 − δ0n)λ2b(1 − p1)n∑

i=1

c2,iP(2)0,n−i(x, t)

+ λ2(1 − b)P (2)0,n (x, t); n ≥ 0, (3)

∂

∂tCm,n(x, t) + ∂

∂xCm,n(x, t)

= −(λ1 + λ2 + ξ + φ(x))Cm,n(x, t)

+ (1 − δm0)λ1

m∑i=1

c1,iCm−i,n(x, t)

+ (1 − δ0n)λ2b

n∑i=1

c2,iCm,n−i(x, t)

+ λ2(1 − b)Cm,n(x, t) + ξCm+1,n(x, t); m ≥ 0, n ≥ 0,

(4)

∂

∂tMm,n(x, t) + ∂

∂xMm,n(x, t)

= −(λ1 + λ2 + ξ + δ(x))Mm,n(x, t)

+ (1 − δm0)λ1

m∑i=1

c1,iMm−i,n(x, t)

+ (1 − δ0n)λ2b

n∑i=1

c2,iMm,n−i(x, t)

+ λ2(1 − b)Mm,n(x, t) + ξMm+1,n(x, t); m ≥ 0, n ≥ 0,

(5)

d

dtQm,n(t) = −(λ1 + λ2 + γ + µ3)Qm,n(t)

+ (1 − δm0)λ1

m∑i=1

c1,iQm−i,n(t)

+ (1 − δ0n)λ2b

n∑i=1

c2,iQm,n−i(t)

+ λ2(1 − b)Qm,n(t) + µ3Qm+1,n(t)

+ α

∫ ∞

0P (1)

m,n(x, t)dx; m ≥ 0, n ≥ 0, (6)

d

dtR0,n(t) = −(λ1 + λ2 + γ )R0,n(t)

+ (1 − δ0n)λ2b

n∑i=1

c2,iR0,n−i(t) + λ2(1 − b)R0,n(t)

+ α

∫ ∞

0P

(2)0,n−1(x, t)dx; m ≥ 0, n ≥ 1, (7)

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d

dtI0,0(t) = −(λ1 + λ2)I0,0(t) +

∫ ∞

0M0,0(x, t)δ(x)dx

+∫ ∞

0V0,0(x, t)β(x)dx + γR0,0(t) (8)

∂

∂tI0,n(x, t) + ∂

∂xI0,n(x, t)

= −(λ1 + λ2 + η(x))I0,n(x, t); n ≥ 1. (9)

The above set of equations are to be solved under thefollowing boundary conditions at x = 0.

I0,n(0, t) = (1 − r){∫ ∞

0M0,n(x, t)δ(x)dx

+∫ ∞

0V0,n(x, t)β(x)dx + γR0,n(t)}; n ≥ 1. (10)

P (1)m,n(0, t) = q

∫ ∞

0P

(1)m+1,n(x, t)µ1(x)dx

+ p

∫ ∞

0P (1)

m,n(x, t)µ1(x)dx + λ1c1,m+1I0,n(t)

+ (1 − δ0n)λ1c1,m+1

∫ ∞

0P

(2)0,n−1(x, t)dx

+∫ ∞

0Mm+1,n(x, t)δ(x)dx

+∫ ∞

0Vm+1,n(x, t)β(x)dx; m ≥ 0, n ≥ 1, (11)

P(1)0,0 (0, t) = q

∫ ∞

0P

(1)1,0 (x, t)µ1(x)dx

+ p

∫ ∞

0P

(1)1,0 (x, t)µ1(x)dx + λ1c1,1I0,0(t)

+∫ ∞

0M1,0(x, t)δ(x)dx

+∫ ∞

0V1,0(x, t)β(x)dx; n = 0, (12)

P(2)0,n (0, t) =

∫ ∞

0I0,n+1(x, t)η(x)dx

+ λ2bc2,n+1I0,0(t) + λ2b

n∑i=1

c2,i

∫ ∞

0I0,n+1−i(x, t)dx

+ r{∫ ∞

0M0,n+1(x, t)δ(x)dx

+∫ ∞

0V0,n+1(x, t)β(x)dx + γR0,n+1(t)}; n ≥ 0,

(13)

V0,n(0, t) = θq

∫ ∞

0P

(1)0,n (x, t)µ1(x)dx

+ θ

∫ ∞

0P

(2)0,n (x, t)µ2(x)dx; m = 0, n ≥ 0, (14)

C0,n(0, t) = (1 − θ )q∫ ∞

0P

(1)0,n (x, t)µ1(x)dx

+ (1 − θ )∫ ∞

0P

(2)0,n (x, t)µ2(x)dx; n ≥ 0, (15)

Mm,n(0, t) =∫ ∞

0Cm,n(x, t)φ(x)dx; m ≥ 0, n ≥ 0.

(16)

We assume that initially there are no customers in thesystem and the server is idle. Then the initial conditionsare,

P (1)m,n(0) = P

(2)0,n (0) = V0,n(0) = Cm,n(0) = Mm,n(0)

= R0(0) = Q0(0) = 0; m ≥ 0, n ≥ 0

I0,n(0) = 0 and I0,0(0) = 1. (17)

The Probability Generating Function(PGF) of this model:

I (x, z2, t) =∞∑

n=1

zn2I0,n(x, t), P (2)

0 (x, z2, t)

=∞∑

n=0

zn2P

(2)0,n (x, t), R0(z2, t) =

∞∑n=0

zn2R0,n(t),

Q(z1, z2, t) =∞∑

m=0

∞∑n=0

zm1 zn

2Qm,n(t),

B(x, z1, z2, t) =∞∑

m=0

∞∑n=0

zm1 zn

2Bm,n(x, t)

where B = P (1), V , C, M ,By taking Laplace transforms from (1) to (17) and solvethe equations, we get,

I 0(x, s, z2) = I 0(0, s, z2)[1 − I (f (a, s))]e−f (a,s)x , (18)

P(1)

(x, s, z1, z2)

= P(1)

(0, s, z1, z2)[1 − B1(f1(s, z1, z2))]e−f1(s,z1,z2)x ,(19)

P(2)0 (x, s, z2) = P

(2)0 (0, s, z2)[1 − B2(f2(s, z2))]e−f2(s,z2)x ,

(20)

V (x, s, z1, z2)

= V (0, s, z1, z2)[1 − V (f5(s, z1, z2))]e−f5(s,z1,z2)x ,(21)

Q(s, z1, z2)

= αP(1)

(0, s, z1, z2)[1 − B1(f1(s, z1, z2))]e−f1(s,z1,z2)x

f3(s, z1, z2),

(22)

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R0(s, z2) = αz2P(2)0 (0, s, z2)[1 − B2(f2(s, z2))]e−f2(s,z2)x

f4(s, z2),

(23)

M(x, s, z1, z2)

= M(0, s, z1, z2)[1 − M(f5(s, z1, z2))]e−f5(s,z1,z2)x ,(24)

C(x, s, z1, z2)

= C(0, s, z1, z2)[1 − C(f5(s, z1, z2))]e−f5(s,z1,z2)x.

(25)

where,

f (a, s) = s + λ1 + λ2,

f1(s, z1, z2) = s + λ1[1 − C1(z1)] + λ2b[1 − C2(z2)]

+ α

[1 − γ

f4(s, z1, z2)

],

f2(s, z2) = s + λ1 + λ2b(1 − p1)[1 − C2(z2)] + α,

f3(s, z2) = s + λ2b[1 − C2(z2)] + γ ,

f4(s, z1, z2) = s + λ1[1 − C1(z1)]

+ λ2b[1 − C2(z2)] + µ3

(1 − 1

z1

)+ γ ,

f5(s, z1, z2) = s + λ1[1 − C1(z1)] + λ2b[1 − C2(z2)].

Simplifying the boundary conditions we can get,

{z1 − (q + pz1)B1(f1(s, z1, z2))}P (1)(0, s, z1, z2)

= λ1C1(z1)I 0(x, s, z2) +− qP

(1)0 (0, s, z2)B1(f1(s, z2))G1(s, z1, z2)

+ P20(0, s, z2)G2(s, z1, z2) (26)

By applying Rouche’s theorem on (26), we get,

qP(1)0 (0, s, z2)B1(f1(s, z2))

=λ1C1(g(z2))I 0(0, s, z2)

[1−I (f (a,s))

f (a,s)

]+ P

(2)0 (0, s, z2)G3(s, g(z2))

G4(s, g(z2)).

(27)

By substituting (27) in required equations we get,

I 0(0, s, z2) = 1 − (s + λ1 + λ2)I 0,0(s)G4(s, z1, z2)

+ (1 − r){ ∫ ∞

0M0(x, s, z2)δ(x)dx

+∫ ∞

0V 0(x, s, z2)β(x)dx + γR0(s, z2)

}(28)

z2P(2)0 (0, s, z2) = I 0(0, s, z2)

[I (f (a, s))

+λ2bC2(z2)

[1 − I (f (a, s))

f (a, s)

]]

+ λ2bC2(z2)I 0,0(s).

+ r{qP

(1)0 (0, s, z2)B1(f1(s, z2))G5(s, z2)

+ P(2)0 (0, s, z2)G6(s, z2)

}(29)

By solving the above equations, we get,

P(1)

(0, s, z1, z2)

=

⎧⎪⎪⎨⎪⎪⎩

I 0(x, s, z2) { λ1C1(z1)G4(s, g(z2)) − λ1C1(g(z2))G1(s, z1, z2) }+ P

(2)0 (0, s, z2){G2(s, z1, z2)G4(s, g(z2))

− G1(s, z1, z2)G3(s, g(z2))}

⎫⎪⎪⎬⎪⎪⎭{

G4(s, g(z2)){z1 − (q + pz1)B1(f1(s, z1, z2))}} ,

(30)

P(2)0 (0, s, z2)

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1 − (s + λ1 + λ2)I 0,0(s))

{G4(s, g(z2))

{I + λ2bC2(z2)

[1 − I

f (a, s)

]

+rλ1C1(g(z2))

[1 − I (f (a, s))

f (a, s)

]G5(s, z2)

}}+ λ2bC2(z2)I 0,0(s)

×{

G4(s, g(z2)) − λ1C1(g(z2))

[1 − I (f (a, s))

f (a, s)

](1 − r)G5(s, z2)

}

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

z2

{G4(s, z2) − λ1C1(g(z2))

[1 − I (f (a, s))

f (a, s)

](1 − r)G5(s, z2)

}

− {G3(s, g(z2))G5(s, z2) + G4(s, g(z2))G6(s, z2) }

×{

r + (1 − r)

(I + λ2bC2(z2)

[1 − I (f (a, s))

f (a, s)

])}

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

(31)

8789


I0(0, s, z2)

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

{(1 − (s + λ1 + λ2)I0,0(s)){z2G4(s, z2)

− r{G3(s, z2)G5(s, z2)

+ G4(s, z2)G6(s, z2)}}+ λ2bC2(z2)I0,0(s){G3(s, g(z2))G5(s, z2)

+ G4(s, g(z2))G6(s, z2) } (1 − r)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

z2

{G4(s, z2) − λ1C1(g(z2))

[1 − I (f (a, s))

f (a, s)

](1 − r)G5(s, z2)

}

− {G3(s, g(z2))G5(s, z2) + G4(s, g(z2))G6(s, z2) }

×{

r + (1 − r)

(I + λ2bC2(z2)

[1 − I (f (a, s))

f (a, s)

])}

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

(32)

V (0, s, z1, z2) = θqP(1)

(0, s, z2)B1(f1(s, z2))

+ θP(2)0 (0, s, z2)B2(f2(s, z2)), (33)

C(0, s, z2) = (1 − θ )qP(1)0 (0, s, z2)B1(f1(s, z2))

+ (1 − θ )P(2)0 (0, s, z2)B2(f2(s, z2)),

(34)

M(0, s, z1, z2) = {(1 − θ )qP(1)0 (0, s, z2)B1(f1(s, z2))

+ (1 − θ )P(2)0 (0, s, z2)

× B2(f2(s, z2))}C(f5(s, z2)) (35)

where,

G1(s, z1, z2)

= {1 − θV (f5(s, z1, z2)) + θV (f5(s, z2))}+ (1 − θ ) { C(f5(s, z2))

× M(f5(s, z2)) − C(f5(s, z1, z2))M(f5(s, z1, z2)) }G2(s, z1, z2)

= θB2(f2(s, z2))[V (f5(s, z1, z2)) − V (f5(s, z2))]

+ (1 − θ )B2(f2(s, z2))

× { C(f5(s, z1, z2))M(f5(s, z1, z2))

− C(f5(s, z2))M(f5(s, z2)) }

+ λ1C1(z1)z2

[1 − B2(f2(s, z2))

f2(s, z2)

]

G3(s, g(z2)) = θB2(f2(s, z2))[V (f5(s, g(z2))) − V (f5(s, z2))]

+ (1 − θ )B2(f2(s, z2))

× { C(f5(s, g(z2)))M(f5(s, g(z2)))

− C(f5(s, z2))M(f5(s, z2)) }

+ λ1C1(g(z2))z2

[1 − B2(f2(s, z2))

f2(s, z2)

]

G4(s, g(z2)) = {1 − θV (f5(s, g(z2))) + θV (f5(s, z2))}+ (1 − θ ){C(f5(s, z2))M(f5(s, z2))

− C(f5(s, g(z2)))M(f5(s, g(z2)))}× (1 − θ + θV (f5(s, z2)))

G5(s, z2) = (1 − θ )C(f5(s, z2))M(f5(s, z2))

+ θV (f5(s, z2))

G6(s, z2) = (1 − θ )B2(f2(s, z2))C(f5(s, z2))M(f5(s, z2))

+ θB2(f2(s, z2))V (f5(s, z2))

+ αγ z2[1 − B2(f2(s, z2))]

f2(s, z2)f3(s, z2).

Theorem 1. The inequality

P (1)(1, 1) + P (2)(0, 1) + Q(1, 1) = ρ < 1,

is a necessary and sufficient condition for the system to bestable, under this condition the marginal PGF of the server’sstate, queue size and orbit size distributions are given by,

I 0(s, z2) = I 0(0, s, z2)

[1 − I (f (a, s))

f (a, s)

], (36)

P(1)

(s, z1, z2) = P(1)

(0, s, z1, z2)

[1 − B1(f1(s, z1, z2))

f1(s, z1, z2)

],

(37)

V (s, z1, z2) = {θqP(1)0 (0, s, z2)B1(f1(s, z2))

+ θP(2)0 (0, s, z2)B2(f2(s, z2))}

×[

1 − V (f5(s, z1, z2))

f5(s, z1, z2)

], (38)

P(2)0 (s, z2) = P

(2)0 (0, s, z2)

[1 − B2(f2(s, z2))

f2(s, z2)

], (39)

C(s, z1, z2) ={

(1 − θ )qP(1)0 (0, s, z2)

×[

1 − B1(f1(s, z2))

f1(s, z2)

]+ (1 − θ ) P

(2)0 (0, s, z2)

×[

1 − B2(f2(s, z2))

f2(s, z2)

] } [1 − C(f5(s, z1, z2))

f5(s, z1, z2)

],

(40)

8790


Q(s, z1, z2) = αP(1)

(0, s, z1, z2)[1 − B1(f1(s, z1, z2))]

f1(s, z1, z2)f4(s, z1, z2),

(41)

R(s, z2) = αz2P(2)0 (0, s, z2)[1 − B2(f2(s, z2))]

f2(s, z2)f3(s, z2), (42)

M(s, z1, z2) ={{

(1 − θ )qP(1)0 (0, s, z2)

×[

1 − B1(f1(s, z2))

f1(s, z2)

]+ (1 − θ ) P

(2)0 (0, s, z2)

×[

1 − B2(f2(s, z2))

f2(s, z2)

] }C(f5(s, z1, z2))

}

×[

1 − M(f5(s, z1, z2))

f5(s, z1, z2)

]. (43)

STEADY STATE ANALYSIS: LIMITINGBEHAVIOUR

By applying the well-known Tauberian property,

lims→0

sf (s) = limt→∞ f (t),

to the above equations, we obtain the steady- state solutionsof this model.

I0(z2) = I0(0, z2)

[1 − I (f (a))

f (a)

], (44)

P (1)(z1, z2) = P (1)(0, z1, z2)

[1 − B1(f1(z1, z2))

f1(z1, z2)

], (45)

V (z1, z2) = {θqP (1)(0, z2)B1(f1(z2))

+ θP(2)0 (0, z2)B2(f2(z2))}

[1 − V (f5(z1, z2))

f5(z1, z2)

], (46)

P(2)0 (z2) = P

(2)0 (0, z2)

[1 − B2(f2(z2))

f2(z2)

], (47)

(48)

C(z1, z2) ={

(1 − θ )qP(1)0 (0, z2)

[1 − B1(f1(z2))

f1(z2)

]

+ (1 − θ ) P(2)0 (0, z2)

[1 − B2(f2(z2))

f2(z2)

] }

×[

1 − C(f5(z1, z2))

f5(z1, z2)

], (49)

Q(z1, z2) = αP (1)(0, z1, z2)[1 − B1(f1(z1, z2))]

f1(z1, z2)f4(z1, z2), (50)

R(z2) = αz2P(2)0 (0, z2)[1 − B2(f2(z2))]

f2(z2)f3(z2), (51)

M(z1, z2) ={{

(1 − θ )qP(1)0 (0, z2)

[1 − B1(f1(z2))

f1(z2)

]

+ (1 − θ ) P(2)0 (0, z2)

[1 − B2(f2(z2))

f2(z2)

] }

× C(f5(z1, z2))} [

1 − M(f5(z1, z2))

f5(z1, z2)

]. (52)

In order to determine I0,0, we use the normalizing condition

P (1)(1, 1) + V (1, 1) + P (2)(1) + Q(1, 1) + R(1)

+ M(1, 1) + C(1, 1) + I0(0, 1) + I0,0 = 1.

For this, let Pq(z1, z2) be the probability generating functionof the queue size irrespective of the state of the system.Then adding equations from (44) to (51), we obtain,

Pq(z1, z2) = P (1)(z1, z2) + V (z1, z2) + P(2)0 (z2)

+ M(z1, z2) + C(z1, z2) + Q(z1, z2) + R(z2), (53)

Pq(z1, z2) = N1(z1, z2)

D1(z1, z2)+ N2(z1, z2)

D2(z1, z2)+ N3(z1, z2)

D3(z1, z2),

where

N1(z1, z2) = I0(0, z2)

[1 − I (f (a))

f (a)

]{G4(z2)f5(z1, z2)

+ λ1C1(g(z2)){1 − θV (f5(z1, z2))

× (1 − θ )[1 − C(f5(z1, z2))M(f2(z1, z2))]},

8791


N2(z1, z2)

= P(2)0 (0, z2)

{G4(z2)

{ [1 − B2(f2(z2))

f2(z2)

]

× (f3(z2) + αz2)f5(z1, z2) + B2(f2(z2))

× f3(z2)[1 − θV (f5(z1, z2))

− (1 − θ )C(f5(z1, z2))M(f5(z1, z2))]}

+ f3(z2)

× (1 − θV (f5(z1, z2))

− (1 − θ )[1 − C(f5(z1, z2))M(f5(z1, z2))]G3(g(z2)))}

,

N3(z1, z2) = P (1)(0, z1, z2)

× { (1 − B1(f1(z1, z2)))[α + f4(z1, z2)] }, (54)

D1(z1, z2) = G4(z2)f5(z1, z2),

D2(z1, z2) = f3(z2)f5(z1, z2)G4(z2),

D3(z1, z2) = f1(z1, z2)f4(z1, z2).

In order to obtain the probability of idle time I0,0, we usethe normalizing condition,

P (1)q (1, 1) + I0,0 = 1.

From which we can have,

I0,0 =(λ1 + α)(λ1 + γ )γ G4(1)( − λ1E(I1)−λ2bE(I2) + α

γ( − λ1E(I1) − λ2bE(I2) + µ3))

Dr,

Dr = (λ1 + α)(λ1 + γ )γ G4(1)( − λ1E(I1) − λ2bE(I2)

+ α

γ( − λ1E(I1) − λ2bE(I2) + µ3))

+ γP(2)0 (0, 1)( − λ1E(I1) − λ2bE(I2)

+ α

γ( − λ1E(I1) − λ2bE(I2) + µ3)) { G4(1)

× {B2(λ1 + α)(λ1 + γ )[(1 − θ )[E(C) + E(M)]

+ θE(V )] + (1 − B2)(λ1 + α + γ )}+ {(λ1 + α)(λ1 + γ )[θE(V )

+ (1 − θ )[E(C) + E(M)]]G3(1)}

+ I0(0, 1)

[1 − I (f (a))

f (a)

]

× { G4(1) + λ1(θE(V )

+ (1 − θ )[E(C) + E(M)])(λ1 + α)(λ1 + γ )γ

− λ1E(I1)

− λ2bE(I2) + α

γ( − λ1E(I1) − λ2bE(I2) + µ3)) }

+ P (1)(0, 1, 1)G4(1){E(B1)

− λ1E(I1) − λ2bE(I2)

+ α

γ( − λ1E(I1) − λ2bE(I2) + µ3))}(λ1 + α)(λ1 + γ ).

STOCHASTIC DECOMPOSITION

Theorem 2. The number of customers in the system understeady state can be decomposed into two independent Prob-ability generating functions, one of which is the PGF ofthe queue size distribution in the priority classical queueand unreliable server with with working breakdown, repair,modified Bernoulli vacation, closedown/startup time, col-lision, orbital search and balking and the other is the PGFof the conditional distribution of the number of customer inthe orbit given that the system is idle. The existence of thestochastic decomposition propertyArtalejo et al. (1994) forour model can be demonstrated easily by showing that

�(z) = (z)�(z) (55)

Proof. The probability generating function (z) of the sys-tem size in the classical priority queue and unreliable serverwith working breakdown, repair, modified Bernoulli vaca-tion, closedown/startup time, collision, orbital search andbalking is given by,

(z) = nr

dr, (56)

nr = { − λ1 − λ2b[1 − C2(z2)]I00} (57)

{ (G2(z1, z2)G4(g(z2)) − G1(z1, z2)G3(g(z2)))f2(z2)f3(z2)

× f5(z1, z2){[1 − B1](f4(z2) + α)}+ (z1 − (q + pz1)B1(f1(z1, z2)))f1(z1, z2)f4(z2)

× { G4(g(z2)){(1 − B2(f2(z2))f5(z1, z2)(f3(z2) + αz2)

+ f2(z2)f3(z2)B2(f2(z2))

× (1 − θV (f5(z1, z2)) − (1 − θ )C(f5(z1, z2))

× M(f5(z1, z2)))} + G3(g(z2))f2(z2)

× f3(z2)(1 − θV (f5(z1, z2)) − (1 − θ )

× C(f5(z1, z2))M(f5(z1, z2))) },dr = {z1 − (q + pz1)}f1(z1, z2)f2(z2)f3(z2)f4(z2)f5(z1, z2),

The probability generating function ψ(z) of the number ofcustomers in the orbit when the system is idle is given by

�(z) = I0,0 + I (0, z2)

I0,0 + I (0, 1)(58)

From the above, we see that �(z) = (z)�(z).

8792


THE AVERAGE QUEUE LENGTH ANDWAITING TIME

The Mean number of customers in the queue and orbit underthe steady state condition is,

Lq1 = d

dz1Pq1 (z1, 1)|z1=1,

Lq2 = d

dz2Pq2 (1, z2)|z2=1. (59)

then,

Lq1 = D′1(1, 1)N ′′

1 (1, 1) − D′′1 (1, 1)N ′

1(1, 1)

2(D′1(1, 1))2

+ D′2(1, 1)N ′′

2 (1, 1) − D′′2 (1, 1)N ′

2(1, 1)

2(D′2(1, 1))2

+ D′3(1, 1)N ′′

3 (1, 1) − D′′3 (1, 1)N ′

3(1, 1)

2(D′3(1, 1))2

,

Lq2 = d ′1(1, 1)n′′

1(1, 1) − d ′′1 (1, 1)n′

1(1, 1)

2(d ′1(1, 1))2

+ d ′2(1, 1)n′′

2(1, 1) − d ′′2 (1, 1)n′

2(1, 1)

2(d ′2(1, 1))2

+ d ′3(1, 1)n′′

3(1, 1) − d ′′3 (1, 1)n′

3(1, 1)

2(d ′3(1, 1))2

,

By Little’s Law, Average waiting time of a customer in thehigh priority queue and low priority orbit is,

Wq1 = Lq1

λ1,Wq2 = Lq2

λ2, (60)

Particular Cases

Case: 1 MX/G/1 Queueing model:If there are no priority arrival, no vacation, no workingbreakdown, no balking, no retrial, no closedown/startuptime and batch arrival. The model under study becomesclassical MX/G/1 queueing system. In this case, the PGFof the busy state is given as,

P (z) = −(1 − B(λ − λC(z)))I0,0

z − B(λ − λC(z))(61)

Case: 2 M/G/1 Queueing model:If there are no priority arrival, no vacation, no workingbreakdown, no balking, no retrial, no closedown/startuptime and single arrival. The model under study becomesclassical M/G/1 queueing system. In this case, the PGFof the busy state is given as,

P (z) = −(1 − B(λ − λz))I0,0

z − B(λ − λz)(62)

The above two results are coincide with the results of Gross.D and Harris. M (1985).

NUMERICAL RESULTS

In order to see the effect of different parameters on thedifferent states of the server we compute some numericalresults. We consider the service time, vacation time andclosedown/startup time to be exponentially distributed tonumerically illustrate the feasibility of our results. By giv-ing the following suitable values for the parameters whichsatisfy the stability condition, we compute the table values.

(λ2, µ1, µ2, µ3, η, θ , α, β, φ, γ , δ, p, p1, r , b)

= (0.1, 7, 7, 0.1, 10, 0.25, 20, 15, 15, 15, 15, 0.2, 0.6, 0.1, 0.8).

(λ1, µ1, µ2, µ3, η, θ , α, β, φ, γ , δ, p, p1, r , b)

= (0.1, 7, 7, 0.3, 11, 0.75, 16, 14, 14, 15, 11, 0.1, 0.7, 0.1, 0.8).

(λ1, λ1, µ3, η, θ , α, β, φ, γ , δ, p, p1, r , b)

= (0.1, 0.1, 0.1, 0.2, 0.1, 20, 0.4, 0.5, 0.2, 1.5, 0.4, 0.4, 0.5, 0.3).

(λ1, λ2, µ1, µ2, µ3, η, θ , α, φ, γ , δ, p, p1, r , b)

= (0.2, 0.2, 10, 10, 0.3, 10, 0.5, 20, 20, 20, 20, 0.4, 0.1, 0.2, 0.8).

Table 1 and 4 clearly shows that as long as the arrivalrate of high priority customers and vacation rate increasesthe servers idle time decreases. Simultaneously the utilisa-tion factor, average queue length for both high priority and

Table 1. Effect of λ1 on various queue characteristics

λ1 I0,0 ρ Lq1 Lq2 Wq1 Wq2

1.4 0.9901 0.0099 0.0238 0.1971 0.0170 1.97131.5 0.9898 0.0102 0.0358 0.5023 0.0238 5.02341.6 0.9896 0.0104 0.0490 0.7579 0.0306 7.57921.7 0.9894 0.0106 0.0635 0.9665 0.0373 9.66501.8 0.9893 0.0107 0.0792 1.1303 0.0440 11.30271.9 0.9892 0.0108 0.0963 1.2511 0.0507 12.51102.0 0.9892 0.0108 0.1147 1.3305 0.0574 13.30552.1 0.9892 0.0108 0.1345 1.3699 0.0641 13.69922.2 0.9892 0.0108 0.1557 1.3703 0.0708 13.7033

Table 2. Effect of λ2 on various queue characteristics

λ2 I0,0 ρ Lq1 Lq2 Wq1 Wq2

2.5 0.9976 0.0024 0.0444 1.5011 0.444 0.60042.6 0.9975 0.0025 0.0444 2.0266 0.444 0.77942.7 0.9975 0.0025 0.0444 2.5832 0.444 0.95672.8 0.9975 0.0025 0.0444 3.1694 0.444 1.13192.9 0.9974 0.0026 0.0444 3.7835 0.444 1.30463.0 0.9974 0.0026 0.0444 4.4240 0.444 1.4747

8793


Table 3. Effect of µ on various queue characteristics

µ I0,0 ρ Lq1 Lq2 Wq1 Wq2

12.1 0.6643 0.3357 0.1069 1.4888 1.0692 14.888012.2 0.6645 0.3355 0.1054 1.3221 1.0545 13.221012.3 0.6646 0.3354 0.1040 1.1561 1.0400 11.560812.4 0.6648 0.3352 0.1026 0.9907 1.0258 9.906912.5 0.6650 0.3350 0.1012 0.8259 1.0119 8.259312.6 0.6651 0.3349 0.0998 0.6618 0.9981 6.617512.7 0.6653 0.3347 0.0985 0.4982 0.9846 4.981512.8 0.6655 0.3345 0.0971 0.3351 0.9713 3.351012.9 0.6656 0.3344 0.0958 0.1726 0.9582 1.725813.0 0.6658 0.3342 0.0945 0.0106 0.9454 0.1056

Table 4. Effect of β on various queue characteristics

β I0,0 ρ Lq1 Lq2 Wq1 Wq2

7.0 0.9992 0.0008 0.0061 0.0326 0.0304 0.32627.5 0.9992 0.0008 0.0061 0.1732 0.0307 1.73238.0 0.9992 0.0008 0.0062 0.3144 0.0309 3.14448.5 0.9992 0.0008 0.0062 0.4561 0.0312 4.56159.0 0.9992 0.0008 0.0063 0.5983 0.0314 5.98269.5 0.9992 0.0008 0.0063 0.7407 0.0316 7.407110.0 0.9992 0.0008 0.0064 0.8834 0.0319 8.834510.5 0.9991 0.0009 0.0064 1.0264 0.0321 10.264411.0 0.9991 0.0009 0.0065 1.1696 0.0324 11.696411.5 0.9991 0.0009 0.0065 1.3130 0.0326 13.1303

low priority customers are also increases. Table 2 revealsthat as long as the arrival rate of low priority customersincreases the servers idle time decreases. Simultaneouslythe utilisation factor, average queue length for low prior-ity customers are increases. Since it is preemptive priorityqueueing system, the arrival of low priority customers doesnot affect the high priority queue. So it remains constant.Table 3 shows that as long as the service rate increases theserver’s idle time increases and the utilisation factor, aver-age queue length for both high priority and low prioritycustomers are decreases.

Graphical Study

We can plot the above data graphically to illustrate thefeasibility of our results.

CONCLUSION

In this paper we have analysed a M [X1], M [X2]/G1, G2/1retrial queue with priority service under modified Bernoullivacation subject to the server working breakdown along

FIGURE 2. Average queue sizes Vs High priority arrivalrate λ1

FIGURE 3. Average queue sizes Vs Low priority arrival rateλ2

with it the close down/start up time also investigated. Inaddition, the effect of impatient behaviour of the customeron a service system is studied. The joint distribution of thenumber of customers in the queue and the number of cus-tomers in the orbit are derived. Numerical examples havebeen carried out to observe the trend of the mean number ofcustomers in the system for varying parametric values. Thispaper analyzes a single-server retrial queue with constantretrial policy, preemptive repeat priority, collisions, orbitalsearch, working breakdowns and repair in order to obtainanalytical expressions for various performance measures ofinterest. The joint steady-state probability generating func-tions of the server state and the number of customers in theorbit are derived. The stochastic decomposition law for thissystem has also been established. Numerical examples havebeen carried out to observe the effects of several parameters

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FIGURE 4. Average queue sizes Vs µ

FIGURE 5. Average queue sizes Vs Vacation rate

on the system. The novelty of this investigation is the dis-cussion of the constant retrial policy, working breakdowns,retrial queueing system with preemptive repeat priorityand collisions of customers. This makes the system morecomplex though realistic.

REFERENCES

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[2] Artalejo. A.R., Falin. J.I. (1994), Stochastic Decompo-sition for Retrial Queues, TOP, Vol. 2, No. 2, 329–342.

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[4] Cheng-Dar Liou (2013), Markovian queue optimisationanalysis with an unreliable server subject to work-ing breakdowns and impatient customers, InternationalJournal of Systems Science Vol. 46, No. 12, 2165–2182.

[5] Cox D.R. (1955), The analysis of non-Markovianstochastic processes by the inclusion of supplementaryvariables, Proc. Cambridge Phil. Soc., Vol. 51, 433–441.

[6] Dong-YuhYang,Ying-YiWu (2017),Analysis of a finite-capacity system with working breakdowns and reten-tion of impatient customers, Journal of ManufacturingSystems, 44, 207–216.

[7] Gross D. and Harris, C.M. (1985), Fundamentals ofQueueing Theory: 2nd Edition, Wiley, New York.

[8] Ioannis Dimitriou and Christos Langaris (2009), AQueueing model with startup/closedown time and retrialcustomers, Taylor and Francis Vol. 25, 248–269.

[9] Jaiswal N.K. (1968), Priority Queues, Academic press,NY.

[10] Kalidass K., Kasturi R. (2012), A queue with work-ing breakdowns, Computers and Industrial Engineering,Vol. 63. 779–783.

[11] Krishna Kumar. B., Vijayalakshmi. G., Krishnamoorthy.A., Sadiq Basha. S. (2010), A single server feedbackretrial queue with collisions, Computers and OperationsResearch, Volume. 37, 1247–1255.

[12] Tao Li and Liyuan Zhang (2017), An M/G/1 RetrialG-Queue with General Retrial Times and WorkingBreakdowns, Math. Comput. Appl., 22, 15.

[13] Yi Peng, Zaiming Liu and Jinbiao Wu (2013), An M/G/1retrial G-queue with preemptive resume priority and col-lisions subject to the server breakdowns and delayedrepairs, J. Appl. Math. Comput.

[14] Zaiming liu andYang Song (2014), The MX/M/1 queuewith working breakdown, Rairo operations research,Vol. 48, 339–413.

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