Quality Technology &
Quantitative Management
Vol. 12, No. 4, pp. 437-464, 2015
QTQM
© ICAQM 2015
An Unreliable Server Retrial Queue with
Two Phases of Service and General Retrial Times
Under Bernoulli Vacation Schedule
Gautam Choudhury1, Lotfi Tadj2,* and Mitali Deka3
1,3
Mathematical Sciences Division, Institute of Advanced Study in Science and Technology,
Paschim Boragaon, India
2
Silberman College of Business, Fairleigh Dickinson University, Vancouver, Canada
(Received October 2013, accepted March 2014)
______________________________________________________________________
Abstract: This paper deals with the steady state behavior of an M/G/1 retrial queue with two successive
phases of service and general retrial times under Bernoulli vacation schedule for an unreliable server.
While the server is working with any phase of service, it may breakdown at any instant and the service
channel will fail for a short interval of time. The primary customers finding the server busy, down, or on
vacation are queued in the orbit in accordance with the FCFS (first come, first served) retrial policy.
After the completion of the second phase of service, the server either goes for a vacation of random
length with probability p or serves the next unit, if any, with probability (1 – p). For this model, we first
obtain the condition under which the system is stable. Then, we derive the system size distribution at a
departure epoch and the probability generating function of the joint distributions of the server state and
orbit size, and prove the decomposition property. We also provide a reliability analysis of this model,
which is one of the chief objectives of the paper. Finally, numerical illustrations show how the
performance measures obtained are used to manage the system.
Keywords: Random breakdown, reliability function, repair time, stationary queue size distribution,
second phase of service.
______________________________________________________________________
1. Introduction
R
etrial queues are characterized by the feature that a customer who finds the server
busy upon arrival is obliged to leave the service area and repeat its request for service
after some time, called “retrial time”. Between trials, the blocked customer joins a group of
unsatisfied customers called “orbit” or ‘retrial group’. Queues in which customers are
allowed to conduct retrials have been widely used to model many practical problems in
telephone switching systems, telecommunication networks, and computers competing to
gain service from a central processing unit. Moreover, retrial queues are also used as
mathematical models for several computer systems such as packet switching networks,
shared bus local area networks operating under the carrier-sense multiple access protocol,
and collision avoidance star local area networks, etc. For a review of the main results and
methods, the reader is referred to the survey papers by Yang and Templeton [64], Falin [30],
Kulkarni and Liang [48] and the book by Falin and Templeton [31]. For more recent
references, see the bibliographical overviews in [6-8, 38]. Further, a comprehensive
comparison between retrial queues and their classical waiting line counterparts can be
found in Artalejo and Falin [10].
*
Corresponding author. E-mail: [email protected]
438
Choudhury, Tadj and Deka
The pioneering studies of retrial queueing systems present the concept of retrial time
as an alternative to the classical models of telephone systems. In this context, each blocked
customer generates a stream of repeated attempts independently of the rest of the
customers in the orbit. Thus, in this type of policy, the intervals between successive
repeated attempts are exponentially distributed with rate n (say), when the number of
customers in the orbit is ‘ n ’. This type of retrial policy is known as the classical retrial policy
and was investigated by Yang and Templeton [64] and Falin [30]. There is a second kind of
policy, called the constant retrial policy, which arises naturally in problems where the server is
required to search for customers (e.g. see [53]) and in communication protocols of type
carrier sense multiple access (CSMA). The latter discipline was introduced by Fayolle [33],
who investigated an M/M/1 retrial queue in which only the customers in the head of the
orbit can request a service after an exponentially distributed retrial time with some
parameter  (say), i.e., the retrial rate is 1   0,n   , where  i , j denotes Kronecker’s
delta, when the number of units in the orbit is ‘n’. Farahmand [32] called this discipline a
retrial queue with FCFS orbit. Choi et al. [14] generalized the constant retrial policy by
considering an M/M/1 retrial queue with general retrial times. Since Fayolle [33], there
has been a rapid growth in the literature, e.g. see [15, 26]. This retrial policy is a useful
device for modeling the retrial phenomenon in communication and computer networks where
repeated attempts are made by processor units independently of the numbers of messages
stored in each node of the network. Besides, this kind of retrial control policy was used for
stability of the ALOHA protocol and unslotted CSMA/CD (carrier sense multiple access with
collision detection) protocol in communication systems.
Retrial queueing systems with general service time and non exponential retrial times
have also received considerable attention during the last decade. The first investigation with
general retrial times was done by Kapyrim [39], who assumed that each customer in the
orbit generates a stream of repeated attempts that are independent of the customers in the
orbit and of the state of the server. However, his methodology was found to be incorrect by
Falin [29]. Subsequently, Yang et al. [65] have developed an approximation method to
obtain the steady state performance for the model of Kapyrim. Later, Gomez-Corral [37]
discussed extensively the M/G/1 retrial queue with FCFS discipline and general retrial
times. In recent years, several retrial models have been analyzed with general retrial times,
details of which may be found in [11, 13, 46, 52, 61].
The classical vacation scheme with Bernoulli service discipline was originated and
significantly developed by Keilson and Servi [42] and co-workers. Kella [43] suggested a
generalized Bernoulli scheme according to which a single server goes on i consecutive
vacations with probability pi if the queue is empty upon his return. At the end of a
vacation period, service begins if a customer is present in the queue. Otherwise, the server
waits for the first customer to arrive.
Recently, there has been considerable attention paid to the study of M/G/1-type
queueing systems with two phases of service under Bernoulli vacation schedule and
different vacation policies, see [19-21, 23], in which after two successive phases of service,
the server may go for a Bernoulli vacation. The motivation for these types of models comes
from some computer networks and telecommunication systems, where messages are
processed in two stages by a single server. As modern telecommunication systems become
more complex and the processing power of microprocessors becomes more expensive, the
advantages of more sophisticated scheduling become more apparent. The need for
scheduling the allocation of resources among two or more heterogeneous types of tasks
arises in many applications.
An Unreliable General Retrial and Two Phases Queue Under Bernoulli Vacation Schedule
439
A wide class of retrial policies for governing the vacation mechanism has also been
discussed in the literature. Most of the analyses for retrial queues concerns the exhaustive
service schedule [5], gated service policy [49], and recently modified vacation policy [13,
41]. A number of papers [40, 44-45, 63] have recently appeared in the queueing literature in
which the concept of general retrial times has been introduced along with Bernoulli
vacation schedule under the FCFS orbit retrial policy. Such types of queueing models occur
in many real life situations where the server may be used for other secondary jobs, for
instance to serve customers in other systems. Allowing the server to take vacations makes
the queueing model more realistic and flexible in studying real-world queueing situations.
Applications arise naturally in call centers with multi-task employees, customized
manufacturing, telecommunication and computer networks, maintenance activities,
production and quality control problems, etc.
The study of queueing models with service interruptions goes back to the 1950s. For
some early papers on service interruptions, we refer the readers to see the papers by Gaver
[36], Avi-Itzhak and Naor [12], Thirurengadan [59], and Mitrany and Avi-Itzhak [51] for
some fundamental works. Li et al. [50], Sengupta [54], Takine and Sengupta [57] and Tang
[58], among others, have studied some queueing systems with interruptions wherein one of
the underlying assumptions is that the service channel undergoes repair instantaneously, as
soon as it fails. On the other hand, retrial queues that take into account servers failures and
repairs were introduced by Aissani [1] and Kulkarni and Choi [47]. As related literature, we
should mention papers such as [2-4]. Wang et al. [62] were the first to study a repairable
M/G/1 retrial queueing model from the viewpoint of reliability. They obtained both the
queueing indices and the reliability characteristics. Recently, Choudhury and Deka [16]
investigated such a repairable M/G/1 retrial queueing model with two phases of service.
Choudhury and Deka [17] and Choudhury et al. [22] further generalized this model to
batch arrival queueing systems. In both papers, they introduced the concept of Bernoulli
admission mechanism in which each blocked customer is admitted to join the retrial group
with a probability ' r ' (say)  0  r  1 , independently of the admission of the rest of
customers arriving in the same batch and/or the actual size of the retrial group. On the
other hand recently, Ke and Chang [40] investigated a similar type of MX/G1 retrial
queueing model with two phases of service, Bernoulli vacation schedule, starting failure,
and general retrial times. In this context, more recently, Choudhury and Ke [18] studied an
MX/G1 retrial queueing system with Bernoulli vacation schedule and general retrial times
for an unreliable server in which the concept of delay period is also introduced. However
no instigation has been done for the unreliable server retrial queueing system with two
phases of service under Bernoulli vacation schedule even in most recent works, although
some aspects have been discussed separately on queueing systems with service interruptions,
two phases of service, Bernoulli schedule vacation, repeated attempts with general retrial
time in the most recent studies. Moreover, another important characteristic for considering
the retrial model with general retrial times is that we always obtain analytical solutions in
closed form expressions. Hence, in this article an attempt is made to study an M/G/1
retrial queue with two phases of service, general retrial times and Bernoulli vacation
schedule for an unreliable server. To this end, the methodology is based on the embedded
Markov chain and the inclusion of supplementary variables approaches.
The rest of the paper is organized as follows. In Section 2, we give a brief description
of the mathematical model. Section 3 deals with the derivation of the stability criteria for
the existence of the stationary regime, and studies the embedded Markov chain describing
the behavior of the system size distribution at a departure epoch. Section 4 deals with the
steady state joint distribution of the server state and the number of customers in the orbit.
440
Choudhury, Tadj and Deka
Some important particular cases of this model are discussed briefly in Section 5. Further,
existence of the stochastic decomposition property is demonstrated in Section 6. Some
important performance measures are derived in Section 7. The reliability analysis of the
model is discussed in Section 8. Numerical illustrations are presented in Section 9 and
Section 10 conclude the paper.
2. The Mathematical Model
We consider an M/G/1 queueing system with two phases of service, where the
primary customers arrive according to a Poisson process with mean arrival rate  . The
server provides each primary customer two phases of heterogeneous service in succession,
a first phase of service (FPS) followed by a second phase of service (SPS) (busy periods).
The service discipline is assumed to be FCFS. The service time random variables Bi ,
i  1, 2 are assumed to follow general laws with probability distribution function (d.f)
Bi ( x ) , i  1, 2, Laplace-Stieltjes Transform (LST) i* ( )  E  e  Bi  , and finite k-th
moments i( k ) , i  1,2 . Here and in the rest of the paper, sub index i  1 (respectively
i  2 ) denotes the FPS (respectively SPS).
While the server is working with any phase of service, it may breakdown at any time
and the service channel will fail for a short interval of time (breakdown periods). The
breakdowns i.e., server’s life times, are generated by an exogenous Poisson process with rates
1 for FPS and  2 for SPS, which we may call some sort of disaster during FPS and SPS
periods, respectively. As soon as a breakdown occurs, the server is sent for repair during
which time it stops providing service to the primary customers till the service channel is
repaired. The customer which was being served just before server breakdown waits for the
server to complete its remaining service. The repair time of the server (denoted by G1 for
FPS and G2 for SPS) random variables are assumed to be arbitrarily distributed with d.f
 G
*
G1 ( y ) and G2 ( y ), LST G1 ( )  E e 1  and G2* ( )  E e  G2  , and finite k-th
(k )
(k )
moments g1 and g2 , respectively. Immediately after the server is repaired, the server is
ready to start its remaining service to customers in either phase of service. In this case, the
service times are cumulative, i.e., we consider preemptive-resume for service time, which
we may refer to as generalized service times. Now if we define H i as the generalized service
time for the i th phase of service (i  1,2) with d.f H i  x  and LST H i* ( )  E  e  H i  ,
then we have
 
n
 ( x ) n  *
H i* ( )    e  x e i x  i
 Gi ( )  dBi ( x )
n 0 0
 n! 
   
*
i (
(2.1)
 ))); for i  1,2.
*
i (1  Gi (
After each SPS service completion, the server may go for a vacation of random length
'V ' (vacation period ) with probability p  0  p  1 or may serve the next unit, if any, with
probability q  1  p , i.e., the server takes a Bernoulli vacation. We assume that the
vacation time random variable ‘V ’ follows a general law of probability with d.f V  y  ,
*
V
LST     E e  , and finite k-th moments   k  . This type of queueing model is
known as a Bernoulli vacation queue with two phases of service and an unreliable server.
The model of this nature, without a second phase of service, was first investigated by Li et
al. [50].
Now to further develop such a type of model, we introduce the concept of repeated
attempts with constant retrial policy, where a primary customer that finds the server busy
with any phase of service, on vacation, or down, leaves the service area and joins a group
An Unreliable General Retrial and Two Phases Queue Under Bernoulli Vacation Schedule
441
of unsatisfied customers, the orbit, in order to seek service again and again until it finds the
server free. The time between successive repeated attempts, i.e., retrial times, Rn ; n  1
of the customers are assumed to be a sequence of i.i.d random variables with d.f C ( x )
*
 R
and LST C ( )  E e  . Finally, we assume that the input process, server’s life time,
server’s delay time, server’s repair time, service time, and retrial time random variables are
mutually independent of each other.
It should be noted here that this type of retrial model for reliable server, i.e., without
service interruption, was also investigated in [41, 44-45]. For example, Ke and Chang [41]
studied an M/G/1 retrial queue with general retrial times and a modified Bernoulli
vacation policy in which, if the orbit is empty, the server takes at most J vacations until
at least one customer is recorded in the orbit when the server returns from a vacation.
Krishnakumar et al. [44-45] dealt with an M/G/1 retrial queue with general retrial times
where the server operates according to a Bernoulli vacation policy as described by Keilson
and Servi [42]. However, our retrial policy assumption varies from theirs. While they
assume that the server always takes a vacation when the orbit is empty, here we assume that
at the vacation completion epoch, the server waits either for the customers, if any, in the
orbit or for new customers to arrive, i.e., the server remains idle, waiting for the customer
from the orbit or new customer arrival.
3. Embedded Markov Chain
We first investigate the necessary and sufficient condition for the stability of the system.
Let tn ; n  Z  be the sequence of epochs at which the n-th service or vacation is
completed, i.e., we consider the epochs at which the total service requested by a customer
expires. Then, the sequence X n  N  t n   , where N  t n  represents the number of
customers in the system at the time instant ' t n ' , forms a Markov chain, which is the
embedded Markov renewal process in our queueing system. We observe that X n ; n  Z 
satisfies the following state equation




X n  X n 1  Wn  U n ,
where U n is the number of units arriving during either n-th completed service or
terminated vacation, and Wn  1 , if the n-th customer in service proceeds from the orbit
and Wn  0 , otherwise.




1
1
1
Theorem 3.1: The inequality 1  1  1 g1(1)   2  1   2 g2(1)  p    C *    is a
necessary and sufficient condition for the system to be stable.


Proof: It is easy to see that X n ; n  Z  is an irreducible and aperiodic Markov chain.
To prove the positive recurrence we may use Foster’s criterion, which states that an
irreducible and aperiodic Markov chain is positive recurrent if there exists a non-negative
function f ( s ) , s  Z  and   0 such that the mean 
drift  s E  f  X n 1  
f  X n  X n  s  is finite for all s  Z  , and  s   for all s  Z  except perhaps at a
finite number. In our case, we take f ( s )  s to obtain








  1 1   g 1   1 1   g 1   p 1 , j  0,
1 1
2
2 2
 1
j  
1
1
1
1
1
1  1  1 g1    2  1   2 g 2    p    C *    ,

where

C *      e   x dC  x  ,
0
j
1,2,...,
442
Choudhury, Tadj and Deka




is the probability that no units arrive during the retrial times. Obviously, 1 1  1 g1 
 21 1   2 g21  p 1  C *    is a sufficient condition for ergodicity. The necessary
condition follows readily from Kaplan’s condition as noted in Sennot et al. [55], namely
 j   for all j  0 and there exists j 0  Z  such that  j  0 for j  j 0 . Further, it
1
1
1
1
1
can be shown that if 1  1  1 g1    2  1   2 g2   p    C *    and Gi   ,
Bi   , V   , and C   for i  1,2 satisfy the regularity conditions, then the system is
stable.
1





Next we assume that X n ; n  Z 
following limiting probabilities


1

is positive recurrent to guarantee that the
 j  lim Pr  X n  j  ,
n 
exist and are positive. Then, the Kolmogorov equations associated with the Markov chain
X n ; n  Z  can be written as:





 j  0C *    qm j  pl j   1  C *      n qm j n  pl j n 
j
n 0
(3.1)
j 1
 C *      n qm j n 1  pl j n 1  ; j  0,
n 1
where

j
m j   m1,i m2, j i represents the probability that ‘j ’ units arrived during the total
i 0
of

both phases of the generalized service time H 1  H 2 ;

mi , j  
e  x   x 
j
j!
0
dH i  x  for i  1,2 represents the probability that ‘j ’ units
arrived during the i th phase of the generalized service time H i ;

j

i 0
0
l j   mi 
e  y   y 
 j  i !
j i
dV  y  represents the probabilities that j units arrived
during the total of the generalized service times plus vacation time H 1  H 2  V .
We now introduce the following probability generating functions (PGFs)




j 0
j 0
j 0
j 0
  z    z j  j , M i  z    z j mi , j , M  z    z j m j , and L  z    z j l j .
Utilizing Equation (2.1) and the convolution property for generating functions, we
have M i  z   H i*     z  for i  1, 2; M  z   M 1  z  M 2  z  , 
and L  z  M  z   *     z  .
j
Multiplying both sides of Equation (3.1) by z and then taking the summation over all
possible values of j  0 , we get on simplification
 z 

 0C *   1  z  q  p *     z   M  z 

 


C *   1  z  q  p *     z  M  z   z 1  q  p *     z  M  z 
.
(3.2)
Now using the normalizing condition  1  1 and l’Hospital’s rule in Equation (3.2),
we get after simplification
An Unreliable General Retrial and Two Phases Queue Under Bernoulli Vacation Schedule
0 



C *     H

C *  
,
443
(3.3)
where  H 
1 1  1 g11  2 1  1 g 21  p 1 and i   i1 for i  1, 2 . The results
of this section can be summarized in the following theorem.
Theorem 3.2: Under the stability condition  H  C *    , the PGF of the embedded
Markov chain, i.e., the PGF of the stationary distribution of the number of units in the
system at a departure epoch, is given by
 z 
C       1  z  q  p     z  M  z 
(3.4)
,
  1  z   q  p     z   M  z   z 1   q  p     z   M  z 
      z    1  G     z           z    1  G     z    .
*
C*
where M  z 

*
H
*
*
1
1
*
*
1
*
2
2
*
2
Remark 3.1: Note that for  i  0 , (i  1, 2) , we get the PGF corresponding to the
stationary distribution of the number of customers in the system at a departure epoch of an
M/G/1 retrial queue with two phases of service and general retrial times under Bernoulli
vacation schedule. A model of similar nature was investigated by Ke and Chang [41].
4. Joint Distribution of the Number in the Orbit and State of the Server
In this section, we obtain the PGF of the joint distribution of the state of the server
and the number in the orbit by treating the elapsed retrial time, the elapsed service time of the
customer, the elapsed vacation time, the elapsed delay time, and the elapsed repair time of the
server as supplementary variables. Assuming the system is in steady state conditions, let
N (t ) be the orbit size (i.e., the number of customers in the retrial group) at time t , R 0 (t ) ,
Bi0 (t ) be the elapsed retrial time and service times for i th phase of service, ( i  1,2 ), of the
customers at time t . In addition, let Di0 (t ) for i  1,2 and V 0 (t ) be the elapsed repair
time of the server during which breakdown occurred at i th phase of service time and
elapsed vacation time of the server at time t , respectively. Further, we introduce the
following random variable:
0, if the server is idle with no customer in the system at time t ,
1, if the server is idle during the retrial time at time t ,

2, if the server is busy with FPS at time t ,

Y (t )  3, if the server is busy with SPS at time t ,
 4, if the server is on vacation at time t ,

5, if the server is under repair during FPS at time t ,

6, if the server is under repair during SPS at time t .
The supplementary variables R 0 (t ) , Bi0 (t ) , V 0 (t ) , and Di0 (t ) for i  1,2 are
introduced in order to obtain a bivariate Markov process N (t ), X (t ) , where X (t )  0 if
Y (t )  0 , X (t )  R 0 (t ) if Y (t )  1 , X (t )  B10 (t ) if Y (t )  2 , X (t )  B20 (t ) if Y (t )  3 ,
X (t )  V 0 (t ) if Y (t )  4 , X (t )  D10 (t ) if Y (t )  5 , and X (t )  D20 (t ) if Y (t )  6 .
Now we define following limiting probabilities:
P0  lim Pr 
X (t ) 0 ,
N (t ) 0,
t 
444
Choudhury, Tadj and Deka
 n (x
) dx lim Pr N
(t ) n, X
(t ) R 0 (t ); x  R 0 (t )  x  dx  , x  0 , n  1 ,
t 


Qn ( y 
) dy lim Pr N
(t ) n, X
(t ) V 0 (t ); y  V 0 (t )  y  dy , y  0 , n  0 .
t 
For i  1, 2


Pi ,n ( x 
) dx lim Pr N
(t ) n, X
(t ) Bi0 (t ); x  Bi0 (t )  x  dx , x  0 , n  0 ,
t 
and for fixed values of x and n  0


Ri ,n ( x , y )dy lim Pr N (t ) n, X (t ) Di0 (t ); y  Di0 (t )  y  dy Bi0 (t ) x , ( x , y )  0 .
t 
Further, it is assumed that C  0   0, C     1, B (0)  0, B () 1, Bi  0   0,
Bi    1, Gi (0)  0, Gi () 1 , that C ( x ) , Bi ( x ) are continuous at x  0 , and that
V ( y ) and Gi ( y ) are continuous at y  0 , respectively, so that
 ( x )dx 
dBi ( x )
dGi ( y )
dC ( x )
dV ( y )
, i ( x )dx 
,
,  ( y )dy 
and i ( y )dy 
1  C (x )
1 V ( y)
1  Bi ( x )
1  Gi ( y )
i  1, 2 are the first order differential (hazard rate) functions of C   , V   , Bi   ,
and Gi   , respectively.
Since the arrival process is a Poisson process, it can be shown from Burke’s theorem
(e.g. see Cooper [24], pp. 187-188) that the steady state probabilities of the bivariate
Markov process N  t  , X  t  exist and are positive recurrent under the same condition as
the steady state probabilities of X n , n  Z  , i.e., if and only if  H  C *    .


4.1. The Steady State Equations
The Kolmogorov forward equations to govern the system under steady state conditions
(e.g. see Cox [25]) can be written as follows:
d
 n ( x )      x   n ( x ) 0 , n  1 ,
dx
d
Qn ( y )      ( y )  Qn ( y )   (1   n,0 ) Qn 1 ( y ) , n  0 ,
dy
(4.1)
(4.2)
and for i  1, 2

d
Pi ,n ( x )      i  i ( x )  Pn ( x )   (1   n,0 ) Pi ,n 1 ( x )   i ( y ) Ri ,n ( x , y )dy , n  0 ,
dx
0
d
Ri ,n ( x , y )     i ( y )  Ri ,n ( x , 
y )  (1   n,0 ) Ri ,n 1 ( x ; y ) , n  0 ,
dy


0
0

 P0   ( y ) Q0 ( y )dy  q  2 ( x ) P2,0 ( x ) dx .
(4.3)
(4.4)
(4.5)
This set of equations are to be solved under the boundary conditions at x  0 :


0
0

 n (0)   ( y ) Qn ( y ) dy  q  2 ( x ) P2,n ( x ) dx , n  1 ,
(4.6)
An Unreliable General Retrial and Two Phases Queue Under Bernoulli Vacation Schedule


0
0
445
P1,n (0) 
 1   n,0    n  x  dx   n,0 P0    ( x ) n 1 ( x ) dx , n  0 ,
(4.7)

P2,n  0    1  x  P1,n  x  dx , n  0 ,
(4.8)
0
and at y  0 ,

Qn (0)  p  2 ( x ) P2,n ( x ) dx , n  0 .
(4.9)
0
Also at y  0 and fixed values of x and for i  1, 2
Ri ,n ( x ,0)   i Pi ,n ( x ) , x  0, n  0 ,
(4.10)
with normalizing condition


 
2  

(4.11)
P0     n  x  dx   Qn ( y )dy      Pi ,n ( x )dx    Ri ,n ( x , y )dxdy  
1.

n 1
i 1 n 0  0
0
0
0 0

4.2. The Model Solution
To solve the system of Equations (4.1)-(4.10), let us define following PGFs for z  1 :




n 0
n 0
n 1
n 1
Q  y; z    z n Qn  y  , Q  0; z    z n Qn  0  ,  ( x , z )   z n n ( x ) ,  (0, z )   z n n (0)
and for i  1, 2


n 0
n 0
Pi ( x , z )   z n Pi ,n ( x ) , Pi (0, z )   z n Pi ,n (0) ,


n 0
n 0
Ri ( x , y; z )   z n Ri ,n ( x ; y ) , Ri ( x ,0; z )   z n Ri ,n ( x ;0) .
Let a (
z )  (1  z ) ; such that a (0)   and a (1)  0 . Proceeding in the usual manner
with Equations (4.1) - (4.5), we get a set of differential equations of Lagrangian type whose
solutions are given by
 ( x ; z )  (0; z )[1  C ( x )]exp{a  0  x } ,
(4.12.1)
Q ( y; z ) Q (0; z )[1  V ( y )]exp{a ( z ) y} ,
(4.12.2)
Pi ( x ; z ) 
Pi (0; z )[1  Bi ( x )]exp{ Ai ( z ) x} , for i  1, 2 ,
(4.12.3)
Ri ( x ; y; z )  Ri ( x ,0; z )[1  Gi ( y )]exp{a ( z ) y} , for i  1, 2 ,
(4.12.4)
and

where Ai ( z )  a ( z )   i 1  Gi*  a  z  

for i  1, 2 .
Again multiplying both sides of Equation (4.6) by z n , then taking the summation over
all possible values of n  0 , and utilizing (4.12.4), for i  1, 2 we get on simplification
Ri ( x ;0; z )   i Pi (0; z )[1  Bi ( x )]exp{ Ai ( z ) x } .
(4.13)
Now multiplying Equation (4.6) by z n , then taking the summation over all possible
values of n  Z  , and utilizing (4.12.1), we get on simplification
446
Choudhury, Tadj and Deka
 (0, z
) qP2 (0, z )  2*  A2  z    Q  0; z   *  a  z     P0 .
(4.14)
Similarly from Equations (4.7)-(4.9), we get


zP1 (0, z )   zP0    0; z  C *     z 1  C *     ,


(4.15)
P2  0; z   P1  0; z )  1*  A1  z   ,
(4.16)
and Q  0; z   pP2  0; z   2*  A2  z   , respectively.
(4.17)
Utilizing (4.16) and (4.17) in (4.15) yields
P1 (0, z ) 
P0 a  z  C
C
*
*
 
  1  z   q  p  a  z      A  z     A  z    z 1   q  p  a  z      A  z     A  z   
*
*
*
1
1
*
2
*
2
*
1
1
2
.
2
(4.18)
Letting z  1 in (4.18), we obtain by l’Hospital’s rule
C *    P0
.
C *     H
(4.19)
 pC *    P0 1  V ( y ) 
,
C *     H
(4.20)
C *    P0 1  Bi ( x ) 
,
C *     H
(4.21)
P (0,1) 
This gives
Q ( y;1) 
and for i  1,2
Pi ( x ,1) 
C *    P0 1  Bi  x   1  Gi ( y ) 
Ri ( x , y;1) 
C *     H
(4.22)
,
Hence from the normalizing condition (4.11), we get
P0 
C *     H
C *  
.
(4.23)
Note that Equation (4.23) represents the steady state probability that the server is idle,
but available in the system. Thus, we summarize our results in the following theorem.
Theorem 4.1: Under the stability condition  H  C *    , the joint distribution of the state
of the server and the size of the orbit has the following partial PGFs
(C
 (x; z ) 
C
*
*
   
   C   1  z   q 
*
H
 
)  z 1  q  p
p
*
*
 a  z     A  z     A  z  1  C  x  exp   x
*
*
1
1
2
2
 a  z      A  z     A  z    z 1   q 
*
*
1
1
2
2
p
*
 a  z      A  z     A  z   
*
1
*
1
2
2
(4.24)
(C      ) a  z  1  B  x  exp  A  z  x 
  1  z   q  p  a  z      A  z     A  z    z 1   q  p  a  z      A  z     A  z   
*
P1 ( x ; z ) 
1
H
C
*
*
*
1
*
1
2
1
*
2
*
1
*
1
2
2
(4.25)
An Unreliable General Retrial and Two Phases Queue Under Bernoulli Vacation Schedule
P2 ( x ; z ) 
C
*
  1  z   q  p
Q ( y; z ) 
C
*
  1  z   q  p
     ) a  z    A  z   1  B  x  exp  A  z  x 
 a  z      A  z     A  z    z  1   q  p  a  z      A  z     A  z   
(C
*
*
*
1
H
*
*
*
1
2
2
*
1
p (C
447
1
*
2
*
2
*
1
1
2
2
(4.26)
     ) a  z    A  z     A  z   1  V  y  exp a  z  y
 a  z      A  z     A  z    z  1   q  p  a  z      A  z     A  z   
*
*
1
H
1
*
2
2
*
1
1
*
2
*
2
*
1
1
2
2
(4.27)
     ) a  z  1  B  x  exp  A  z  x   1  G  y  exp  a  z  y
  1  z   q  p  a  z      A  z     A  z    z 1   q  p  a  z      A  z     A  z   
 1 (C
R1 ( x , y ; z ) 
C
*
*
1
H
*
*
1
*
2
*
2
*
1
1
2
2
(4.28)
 (C      ) a  z    A  z   1  B  x  exp  A  z  x   1  G  y   exp  a  z  y
  1  z   q  p  a  z      A  z     A  z    z 1   q  p  a  z      A  z     A  z   
*
2
C
1
*
1
*
R2 ( x , y ; z ) 
2
*
1
H
1
*
2
*
2
2
*
1
1
*
2
*
2
*
1

where a (
z )  (1  z ) and Ai ( z )  a ( z )   i 1  Gi*  a  z  

1
2
2
(4.29)
for i  1, 2 .
Next we are interested in investigating the marginal orbit size distribution due to
system state of the server.
Theorem 4.2: Under the stability condition  H  C *    , the marginal PGFs of the
server’s state orbit size distribution are given by
(C
 (z ) 
C
P1 ( z ) 
P2 ( z ) 
*
*
 
 a  z  
   
   C   1  z   q 
*
p
*
H
) z 1  q  p
A1
 z  C   1  z   q 
*
A2
 z  C    1  z   q 
*
*
2
   
2
2
p
*
 a  z      A  z     A  z   
*
1
*
1
2
2
(4.30)
) a  z  1    A  z   
*
H
1
1
*
1
p
*
2
*
1
*
(C
*
*
1
 a  z      A  z     A  z    z  1   q  p  a  z      A  z     A  z   
*
p
*
1
 A  z     A  z    z 1   q 
*
1
(C
*
 a  z     A  z     A  z  1  C   
*
*
*
1
   
*
2
)a  z  1
*
H
1
1
2
2
(4.31)
 A  z   1    A  z  
1
2
2
*
1
*
1
*
 a  z      A  z     A  z    z 1   q 
*
*
2
2
2
p
*
 a  z      A  z     A  z   
*
1
*
1
2
2
(4.32)
p (C
Q(z) 
R1 ( z ) 
C
*
*
   
A1
H
) 1
 A  z     A  z   1    a  z   
*
1
*
2
2
  1  z   q  p  a  z      A  z     A  z    z 1   q  p  a  z      A  z     A  z   
*
*
*
1
(C
*
*
 z  C   1  z   q 
*
p
*
*
1
*
2
   
H
*
2

*
) 1  1
1
*
1
2
2
(4.33)
 A  z  1  G  a  z  
*
1
1
 a  z      A  z     A  z    z  1   q  p  a  z      A  z     A  z   
*
1
*
1
2
*
2
*
1
*
1
2
2
(4.34)
R2 ( z ) 
(C
*
A2
 z  C    1  z   q 
*
p
*
*
   
H
)   A1
 z   1    A  z  1  G  a  z  
*
2
*
2
2
 a  z      A  z     A  z    z  1   q  p  a  z      A  z     A  z    
*
1
*
1
2
*
2
*
1
*
1
2
2
(4.35)
Proof: Integrating expressions (4.24) - (4.26) with respect to x and using the well-known
result of renewal theory
1  f * ( s ) 


,
 sx
e
1
F
(
x
)
dx


(4.36)



s
0
448
Choudhury, Tadj and Deka
where F   is the d.f of a random variable and f   represents LST of it, we get
formulae (4.30) - (4.32), respectively. Similarly, integrating (4.27) with respect to y , we get
formula (4.33). Again, integrating Equations (4.28) and (4.29) with respect to y , we get
*

 i Pi ( x ; z ) 1  Gi*  a ( z )  
0
a(z )

Ri ( x ; z ) 
Ri ( x , y; z )dy
, for i  1, 2.
(4.37)
Further, integrating (4.37) with respect to x for i  1,2 , we claimed in formula (4.34)
and (4.35), respectively. Our next objective is to investigate the distribution of the number
of customers in the orbit.
Corollary 4.1: Let  j be the stationary distribution of the number of customers
in the orbit

j
under the steady state condition, then its corresponding PGF   z  
 z  j is given by
j 0
z 
*
*
*
C    1  z  q  p  a  z   1


     ) 1  z 
 A  z     A  z    z  1   q  p  a  z      A  z     A  z   
(C
*
H
*
1
*
2
*
2
1
*
1
2
2
(4.38)
Proof: The result follows directly from Theorem 3.2. With the help of the PGFs   z  ,
Q  z  , Pi  z  , Ri  z  for i  1, 2 and P0 , we get the distribution of the number of
customers in the orbit having PGF
  z  P0    z   P1  z   P2  z   Q  z   R1  z   R2  z  .
By direct calculation we can obtain (4.38).
Remark 4.1: If we take 

0 (i.e., there is no breakdown in the system) and
1
2
*
*
for
and
 H*   11   21  p  2   1 , then formula (4.38)
i  Ai  z    i  a  z  
i  1,2;
reduces to


(C      H ) 1  z 
z 
*
*
*
*
*
*
*
C   1  z  q  p  a  z   1  a  z    2  a  z    z 1  q  p  a  z   1  a  z    2  a  z  
*

*

 


,
which is consistent with expression (48) of Ke and Chang [40] for   1 (i.e., there is no
failure in the system) and Pr  X 1 1 (i.e., for unit arrival case) in their paper.
5. Some Particular Cases
In this section we discuss briefly some particular cases of our model, which are
consistent with the existing literature. Suppose the retrial time distribution is exponential
1
1  exp  x  ; x  0 , then C * ( )        , and therefore Equations
with d.f C ( x ) 
(4.19) and (3.4) yield
 1  z 1   
,
z  
*
q  p  a  z   1*  A1  z    2*  A2  z    z     

(5.1)

and
 z 
 1  z 1    q  p *  a  z    1*  A1  z    2*  A2  z  
where

  H      / .
q  p  a  z     A  z    A  z    z     
*
*
1
1
*
2
2
,
(5.2)
An Unreliable General Retrial and Two Phases Queue Under Bernoulli Vacation Schedule
449
These are the PGFs of the orbit size and system size distributions of the unreliable
M/G/1 retrial queue with two phases of service and constant repeated attempts under
Bernoulli vacation schedule. Note that for p  0 (i.e., there is no Bernoulli vacation in
the system) and  i  0 (i.e., there is no breakdown in the system) for i  1,2 with
 2*  A2  z    1 , the above expression (5.2) is consistent with expression (2.13) of Choi et al.
[14].
Suppose that there is no retrial time in the system, then with C *     1 , Equation (3.4)
yields
1   H 1  z  q  p *  a  z   1*  A1  z    2*  A2  z  
 q  p *  a  z   1*  A1  z   2*  A2  z    z
  0  z  , (say),
 z 
(5.3)
which is the PGF of the system size distribution of an M/G/1 unreliable queue with two
phases of service and a single vacation under Bernoulli vacation schedule. Note that for
 i  0 (i  1,2) , the above expression (5.3) is consistent with the result obtained by
Choudhury and Madan [20], (e.g., see Section 3) for the single unit arrival case. Again if
we take z   *  a ( z )  in Equation (3.4), then we get on simplification
 z 
 qC
     H  1  z  q  pz  1*  A1  z   2*  A2  z  
,
*
C *   1  z  q  pz  1*  A1  z    2*  A2  z    z 1   q  pz  1*  A1  z    2*  A2  z  

1
*


1

(5.4)
where  H  1 1  1 g1  2 1   2 g 2 . This is the PGF of the number of customers in
the system at a service completion epoch of the unreliable M/G/1 retrial queue with two
phases of service, Bernoulli feedback mechanism and generalized repeated attempts.
Further, if we take p  0 (i.e., there is no Bernoulli feedback in the system), then
expression (5.4) yields
C
     H  1  z  1*  A1  z   2*  A2  z  
,
 z 
*
C *   1  z  1*  A1  z    2*  A2  z    z 1  1*  A1  z    2*  A2  z  

1

*

1
(5.5)

where  H  1 1  1 g1  2 1   2 g 2 . This is the PGF of the system size distribution
of the unreliable M/G/1 retrial queue with two phases of service and generalized repeated
attempts. Note that for 1  0 with  2*  A2  z    1 , the above expression (5.5) is
consistent with expression (16) of Gomez-Corral [37]. It should be noted that a model of
similar nature has also been investigated by Wang and Li [61].
6. Stochastic Decomposition
In this section we study the stochastic decomposition property of the system size
distribution of our model. The literature on vacation models recognizes this property as
one of the most interesting features in this mater, e.g., see Doshi [27], and Fuhrmann and
Cooper [34]. Stochastic decomposition for retrial models is also found in Yang et al. [65]
and Yang and Templeton [64]. The existence of the stochastic decomposition property for
our model can be demonstrated easily by showing that
 z   0 z   z  ,
(6.1)
450
Choudhury, Tadj and Deka
where  0  z  is the PGF of the system size distribution of an M/G/1 queue with
unreliable server and delaying repair under Bernoulli vacation schedule given by formula
(5.3) and   z  is the PGF of the conditional distribution of the number of customers in
the orbit given that the system is idle 
 P0    z   /  P0   1  which is equal to
 z 
C
1    C   1  z   q 
*
H
p
   q  p  a  z      A  z     A  z    z 
 a  z      A  z     A  z    z  1   q  p  a  z    
   
*
*
*
*
*
1
H
*
1
2
*
1
1
2
*
2
2
*
1
 A  z     A  z   
*
1
2
2
More specifically, we may call it the additional system size distribution due to retrial time.
The expected number of customers in the orbit during the retrial time, i.e., the mean of
the additional system size due to retrials is found to be
L    1




H 1  C *   

C *     H


   2  1   g 1 
1 1
 1
 

2

2
1
  2  1   2 g 2 



2
1
2
1
2
 1 1  g1    2  2  g 2   1  C *    
2 1   H  C *      H




 2 p   2   2 1 11 1  1 g11   21 1   2 g 21  1  C *    


*
2 1   H  C      H
1 
1
  
2

2
1
2

1   g  1   g  1  C
1    C      
1
1 1
2
1
2
*
H
*

   
.
H
Remarks 6.1: From expression (6.1), we observe that the system size distribution at a
departure epoch of the unreliable M/G/1 retrial queue with generalized repeated attempts
under Bernoulli vacation schedule can be decomposed into distributions of two
independent random variables viz.-
(i)
The system size distribution of an M/G/1 queue with two phases of service and
unreliable server under Bernoulli vacation schedule [represented by the first term of
the right hand side of expression (6.1)] and
(ii) The additional system size distribution due to retrials [represented by the second term
of the right hand side of expression (6.1)].
This confirms the Stochastic Decomposition Property of Fuhrmann and Cooper [34].
Further, we note that similar types of decomposition have been presented by Ke and Chang
[41] for a reliable server service system.
7. System Performance Measures
Our next objective is to provide explicit expressions for the system state probabilities
and some important performance measures of the system. First of all, we derive the system
state probabilities and results are summarized in the following theorem.
Theorem 7.1: If the system is in steady state conditions, then we have
(i)
The probability that the server is idle and the system is empty is
PE  1   H / C *    .
(ii) The probability that the server is idle but the system is nonempty is

PNE  H 1   H / C *    .


.
An Unreliable General Retrial and Two Phases Queue Under Bernoulli Vacation Schedule
(iii) The probability that the server is busy with FPS
451
1
is PB1  1  .
1
(iv) The probability that the server is busy with SPS is PB2   2  .
1
1
(v) The probability that the server is under repair during FPS is PR1  11  g1  .
1
1
(vi) The probability that the server is under repair during SPS is PR2   2  2  g2  .
(vii) The probability that the server is on vacation is PV  p   .
 ( z ), PV  lim Q ( z ) and for i  1,2 , PBi  lim Pi ( z ) , P 
Proof: Note that PNE  lim
Ri
z 1
z 1
z 1
lim Ri ( z ) , and P 
1 P  P  P  P  P  P . The stated formulae follow by
1
z 1
direct calculation.

E
NE
V
B1
B2
R1
R2

Secondly, we derive the mean orbit size and the mean system size of this model under
stability conditions.
Theorem 7.2: Let LO and LS be the expected number of units in the orbit and system,
respectively, then under the steady state condition we have
 H 1  C *    

LO
C


*
    H 


 2  1 2  1  1 g11


 



 2 p   2   2 1 11 1  1 g11   21 1   2 g21  1  C *    


2 C

2


2


C
    H 
    H 
1
2
1
2 
 1 1  g1    2  2  g2   1  C *    

*
2 C     H
2
1
  2  1   2 g2 
(7.1)


 2 11  21 1  1 g11 1   2 g 21 1  C *    
*
*
,
and
LS  LO   H  L  LM /G1 ,G2 (UR )/,1( BS )/SV ,
(7.2)
where
LM /G1 ,G2 UR /1( BS )/
H 
SV



 2  1(2) 1  1 g11


2

  2(2) 1   2 g2 
2 1   H 
1
 

2
2 
 p   



 2 1 1(1) g1 2    2  2(1) g 2 2   p 1 1 1  1 g11  2 1   2 g 2



2 1   H 
 2 1 2 1  1 g11 1   2 g21
1   H 
1

,
is the mean system size of an M/G/1 unreliable queue with two phases of service and
single vacation under Bernoulli vacation schedule.
Proof: The results follow directly by differentiating (4.19) and (3.4) with respect to z, then
taking the limit z  1 and using l’Hospital’s rule.
Remark 7.1: Once we have obtained the expected number of units in the orbit, it is trivial
to obtain the other related performance measure viz. mean waiting time. The mean waiting
452
Choudhury, Tadj and Deka
time W in the steady state is obtained with the help of Little’s formula LO  W .
Next we derive the mean busy period and expected length of a busy cycle under the
steady state condition.
Theorem 7.3: Let Tb and Tc be the length of a busy period and length of a busy cycle,
respectively, then under the steady state conditions, we have

E (Tb )



1 1  1 g11   2 1   2 g21
C *     H

1
p  
,
C *     H
(7.3)
.
(7.4)
and
E (Tc ) 
C *  
 C *      H 
Proof: The results follow directly by applying the argument of an alternating renewal
process, which lead to the well-known result
E (Tb ) 

1 1
 1 ,

  0

(7.5)
and
E (Tc )    0  .
1
(7.6)
Inserting (3.3) in (7.5) and (7.6), we get (7.3) and (7.4), respectively.
8. Reliability Analysis of the Model
In this section, firstly we will consider two reliability indices of the system viz. the
system availability and failure frequency under the steady state conditions. Let AV (t ) be
the system availability at time ‘t ’, that is, the probability that the server is either working for
a customer or in an idle period. The steady state availability of the server will be
AV  lim AV (t ) .
t 
Theorem 8.1: The availability of the server and failure frequency of the server under the
steady state condition are respectively given by
AV  1 




11 1  C *      1 g11   2 1  C *       2 g21   p (1)
C
*
 
,
(8.1)
and
1
1

M f 11    2  2  .
Proof: The result follows directly by considering the following equations


0
0
AV 
P0   P1  x ,1 dx   P2  x ,1 dx ,


0
0

M f 1  P1  x ;1 dx   2  P2  x ,1 dx .
(8.2)
An Unreliable General Retrial and Two Phases Queue Under Bernoulli Vacation Schedule


1
Now since 0 1  Bi ( x )  dx  0 xdBi ( x )  i
(4.21), we get (8.1) and (8.2) respectively.
453
for i  1, 2 ; therefore from Equations
Next, we are interested in the reliability function of this model. To obtain it, we denote
by  the time to the first failure of the server, then the reliability function of the server is
defined as R
 t  Pr   t  .
Theorem 8.2: Let R *   be the Laplace transform of the reliability function R  t  ; then
we have
*  
*  
1
1  *0     1
1  C *       2
1  1*   1  




















1
R *  


3* 1 ; 
   2 
1      2   


*
2
*4 1 , 2 ; 

1  

*
(8.3)
   ,
where
  
0
*0   
,
   0    C *    

1*

     0     A*      0    C *       0   
  
   
0
   C *               C *      A*  
,
     1  0   
,
*2   
   0            C *     A*   




     1  0    1* 1   
,
3* 1 ;  
   0            C *     A*   

*4



 p     1  0    1* 1     2*  2   
,
1 , 2 ;  
   0             C *      A*  
 0      0   , A *   
 q  p *    1*   1  2*    2 ,
and 0   is the unique root of the equation
x


1
  x       x  C *       A *  ; x  ,

  

and A *  ; x   q  p *      x  1*   1     x   2*    2     x  .
Proof: In order to find the reliability function of the server, let us assume that the failure of
the server be absorbing states. Then we obtain a new system. In this new system, we use the
same notation as in the previous section. Then we get the following set of equations:


d
P0  t  
 P0  t     y  Q0  y; t  dy  q  2  x  P2,0  x ; t  dx ,
dt
0
0
(8.4)
454
Choudhury, Tadj and Deka
 

   n  x ; t       x   n  x ; t  0, n  1 ,


t
x

(8.5)

 
t   1   n,0  Qn 1  y; t  , n  0 ,
   Qn  y; t       y   Qn  y;
 t y 
(8.6)
and for i  1,2 ,
 

t   1   n,0  Pi ,n  x ; t  , n  0 .
 
 Pi ,n  x ; t      i  i  x   Pi ,n  x ;
 t x 
(8.7)
These equations are to be solved under the following boundary conditions:


0
0

 n  0; t     y  Qn  y; t  dy  q  2  x  P2,n  x ; t  dx , n  1 ,
(8.8)


 


P
0;
t


P
t
1


x
;
t
dx









1,n
n ,0
n
 n,0 0
  0   x  n  x ; t  dx , n  0 ,
0



P2,n  0; t    1  x  P1,n  x ; t  dx , n  0 ,
(8.9)
(8.10)
0

Qn  0; t   p  2  x  P2,n  x ; t  dx , n  0 ,
(8.11)
0
and with initial condition P0  0   1 . Now, taking the Laplace transform of these equations,
we get


0
0
 P0* 
  1  P0*       y  Q0*  y;  dy q  2  x  P2,0  x ;  dx ,
       x   n*  x ;   x  n*  x ; , n  1 ,
(8.12)
(8.13)

 1   n,0  Qn*1  y;  , n  0 ,
       y   Qn*  y;   y Qn*  y;  
(8.14)
and for i  1, 2 ,
     i  x   Pi *,n  x ;   x Pi *,n  x ;   1   n,0  Pi *,n 1  x ;  ,
n0.
(8.15)
Similarly, we have


0
0

 n*  0;     y Qn*1  y;  dy  q  2  x  P2,*n  x ;  dx , n  1 ,


 

P1,*n  0;    n,0 P0*    1   n,0    n*  x ;  dx      x  n*  x ;  dx , n  0 ,
0

 0

P2,*n  0;    1  x  P1,*n  x ;  dx , n  0 ,
0
(8.16)
(8.17)
(8.18)
An Unreliable General Retrial and Two Phases Queue Under Bernoulli Vacation Schedule

Qn*  0;   p  2  x  P2,*n  x ;  dx , n  0 .
455
(8.19)
0
We introduce the following generating functions for z  1 :


n 1
n 1


n 0
n 0
 *  x , ; z    z n n*  x ;  ,  *  0, ; z    z n n*  0;  ,
Q *  y, ; z    z n n*  y;  , Q *  0, ; z    z n Qn*  0;  ,
and for i  1, 2,


n 0
n 0
Pi *  x , ; z    z n Pi *,n  x ;  and Pi *  x , ; z    z n Pi *,n  x ;  .
Now from Equations (8.13) to (8.15), we have
 *
 x , ; z   *  0, ; z  1  C  x  exp       x  ,

(8.20)

Q *
 y, ; z  Q *  0, ; z  1  V  y   exp     z     y ,
(8.21)
and for i  1, 2,


Pi * 
 x , ; z  Pi *  0, ; z  1  Bi  x   exp     z   i    x ,
(8.22)
where  
z   1  z  is defined in Section 4. Similarly, from Equations (8.16) to (8.19),
we get

Q *  0, ; z  pP2*  0, ; z   2*    2    z   ,
(8.23)

P2*  0, ; z  P1*  0, ; z  1*    2    z   ,
(8.24)
P1*  0,
; z   P0*   
1
 z      z   C *      *  0, ; z  ,
z    


(8.25)
and
*

 0, ; z  P1*  0, ; z  q  p *     z    1*   1    z    2*    2    z  
 1       P0*   .
(8.26)
Now utilizing (8.26) in (8.25), we get
*
 *  0, ; z        P0*    1
P1*  0, ; z  
,
A *  ; z 


where A*  ; z   q  p *     z   1*   1    z    2*    2    z   , from which, we
get on simplification
 *  0, ; z  
z          A *  ; z   P0*    z    
  z      z  C *      A *  ; z   z    


Consider the denominator of expression (8.27), i.e.
.
(8.27)
456
Choudhury, Tadj and Deka
f (z ) 
 z      z  C
*
     A*    z     .
It can be shown that the function f  z  is convex. Hence by Rouche’s theorem,
f  z  has exactly one root 0   inside the unit circle z  1 for Re  z  , and therefore
we have
P0*  


0  
1 
1 
,
        0    C *     

(8.28)

where  0      0   and 0   is the unique root of the equation
x

1
 x      x   C *       A *  ; x  .

  
Now utilizing (8.28) and (8.27) in Equation (8.20), we get


0  
z      A *  ; z  1 
  z    

  0    C *      



 *  x , ; z  
 z      z   C *     A *  ; z   z     






 exp       x  1  C  x   .
Hence we have

 *  ; z     *  x , ; z  dx
0




0  
 z      A *  ; z  1 
 1  C *      


z





*



   0    C      



.
  z      z   C *     A *  ; z   z           








Similarly, utilizing (8.28), (8.27) and (8.25) in Equation (8.22), we get on simplification
P1*  x , ; z  
     0    z  1  B1  x   exp     z   1    x 


   0       z      z   C *     A *  ; z   z      


,
(8.29)
and therefore we have

P1*  ; z    P1*  x , ; z  dx
0

     0    z  1  1*    z   1    


   z   1       0       z      z  C *     A *  ; z   z      


.
Similarly, utilizing (8.29), (8.24) and (8.23) in Equations (8.22) and (8.21) and then
integrating, we get on simplification
An Unreliable General Retrial and Two Phases Queue Under Bernoulli Vacation Schedule
457

P2*  ; z    P2*  x , ; z  dx
0

     0    z  1*    z   1    1   2*    z    2    


   z    2       0       z      z  C *     A *  ; z   z      


,
and

Q *  ; z    Q *  y, ; z  dy
0

 p     0    z  1*    z   1     2*    z    2    1   *    z     


   z        0       z      z   C *     A *  ; z   z      


.
Hence, from the above expressions, we have
R *   P0*     *  ;1  P1*  ;1  P2*  ;1  Q *  ;1 .
By substitution, we obtain formula (8.3).
Corollary 8.1: The mean time to the first failure (MTFF) of the server is given by
MTTFF
C
*
      0  1  C    
C

1  C
*
0
*
   1    0  
*
      0              0    C    1    0  
*
0
C
0
1    0   
0

2
1  
*
1
1
*
*

1
1
*
2
0
*
0
1
0
*
1
2
(8.30)
2
*
1
 1 2 1   0  0   1   1   1   2   2 
*
2
   1      1          
          1      
1
*
1
*

2
2
.
Proof: From the above expression (8.3) we obtain (8.30) since MTTF  R *  
 0
.
9. Numerical Illustrations
In this section, we present a possible application and some numerical examples to
explain that our model can represent the application reasonably well. In wireless network,
access point is used to connect between wireless and wired networks. Wireless access point
provides services essentially in two phases viz. FPS (first phase of service) followed by SPS
(second phase of service) parallel to a WLAN. It serves as an interface point and bridges
between wireless devices and wired network, so the wireless devices can access the
resources of the wired network. It further provides wireless linkage between wireless
devices that may be out of range of each other. Typically, wireless device’s connection
requests arrive at the access point following the Poisson stream. When requests arrive at the
access point, one request is selected to be served and the rest of the requests join the buffer
and retry again after a random period of time. In the scan to keep the access point
functioning well or serve the next connection request. In addition, in practice, the access
point may fail and can be repaired. The repair time depends on the degree of failure of
access point. Because the system performance may be heavily affected by access point
breakdown, it is well worth to investigate such system from the queueing theory viewpoint
to develop a proper management policy. In this scenario, access point, buffer in the access
point, retransmission policy, and maintenance activities correspond to the server, the orbit,
the retrial discipline, and the vacation policy, respectively, in the queueing terminology.
458
Choudhury, Tadj and Deka
The results obtained in this paper are easily implementable and can be used to manage
this queueing system efficiently. We illustrate in this section by considering two applications,
namely, the optimal design and the cost effectiveness analysis.
9.1. Optimal Design
The optimal design of a queueing system is to determine the optimal system
parameters using some cost function. To illustrate, let c h be the holding cost per unit time
for each customer present in the system, c o be the cost per unit time for keeping the server
on and in operation, c s be the setup cost per busy cycle, and c a be the startup cost per
unit time for the preparatory work of the server before starting the service. These unit costs
can be combined with the performance measures obtained above to write the system total
expected cost per unit of time as
TC c h Ls  c o
E (Tb )
E (Ti )
1
 ca
 cs
 c h Ls  c o  (c a   c s  c o ) 0 .
E (Tc )
E (Tc )
E (Tc )
Here E (Ti )  1  is the expected length of an idle period, while E (Tb ) and E (Tc )
are the expected length of a busy period and a busy cycle, and are given by (7.3) and (7.4),
respectively. To show how any parameter can affect the system performance, let us assume
that the various distributions involved are exponential. Also, consider the following values
for the system parameters: p  0.5 ,   0.25 , c h  5 , c o  100 , c s  1000 , c a  100 ,
1  0.5 ,  2  0.5 , 1(1)  0.5 ,  2(1)  0.4 , g1(1)  1(1) 5 , g2(1)   2(1) 5 ,  (1) 
( 1(1)   2(1) ) 4 , and mean inter-retrial time 1/  0.1 . In these circumstances, the system
total expected cost per unit of time is TC  281.31 . To investigate the effect of retrials on
the system performance, we give different values to the mean inter-arrival rate and record
the corresponding value of the system total expected cost. Figure 1 below shows the
variations of the total expected cost. We find that the minimum cost is TC  269.76 , and it
is attained when the mean inter-retrial time is 8.8.
Figure 1: Effect of the mean inter-retrial time on the system total
expected cost per unit of time.
Suppose now we are interested in the effect of the Bernoulli vacation schedule on
system performance. Keeping the values of the system parameters as above, we vary
probability of a vacation from 0 to 1 and again record the corresponding values of
system total expected cost. Figure 2 below depicts the variations of the cost with
the
the
the
the
An Unreliable General Retrial and Two Phases Queue Under Bernoulli Vacation Schedule
459
probability of a vacation. We see that the higher the probability of a vacation, the lower the
system total cost. The lowest cost is TC  274.25 , reached when p  1 . In managerial
terms, this means that the server should always take a vacation at the completion of the
second phase of service.
Figure 2: Effect of the Bernoulli vacation on the system total expected
cost per unit of time when the mean inter-retrial time is 0.1.
Finally, we want to see the effect of the Bernoulli vacation probability when all the
data are kept the same, except for the mean inter-retrial time which is given the value 8.8.
Figure 3 shows the variations of the system total cost when the probability of a vacation
varies from 0 to 1. The lowest cost is TC  160.99 , reached when p  0.5 . In managerial
terms, this means that half of the time, the server should take a vacation at the completion
of the second phase of service. Investigation of the effect of other system parameters can
be conducted along the same lines.
Figure 3: Effect of the Bernoulli vacation on the system total expected
cost per unit of time when the mean inter-retrial time is 8.8.
9.2. Cost-effectiveness Analysis
The previous approach did not involve the server availability A . To include this
460
Choudhury, Tadj and Deka
index, we propose the following cost-effectiveness analysis (CEA). CEA is a form of
economic analysis that compares the relative costs and outcomes (effects) of two or more
courses of action. CEA was developed in the military and is often used in health care
services where it may be inappropriate to monetize health effect, to compare the relative
value of various clinical strategies. Typically the CEA is expressed in terms of a
cost-effectiveness ratio (CER ) , where the denominator is a gain in health from a measure
and the numerator is the cost associated with the health gain.
CER 
cost new strategy  cost current practice
effect new strategy  effect current practice
.
The result might be considered as the "price" of the additional outcome purchased by
switching from current practice to the new strategy. If the price is low enough, the new
strategy is considered "cost-effective." To apply this concept to our service system, we use
for cost the system total expected cost and for effect the server avilability. Suppose we are
interested in investigating the necessity of a server vacation following a second phase of
service. We will assume that the current practice is that the server never takes a vacation
(i.e., p  0 ) and we want the new strategy to have the server always take a vacation at the
end of a second phase of service (i.e., p  1 ). Table 1 below shows data and calculations
for this case.
Table 1. Cost-effectiveness ratios.
Bernoulli Vacation Probability
0
1
Total Expected Cost
288.366
274.250
Server Availability
0.973
0.916
ACER
296.26
MCER
244.82
Table 2. Effect of the vacation probability on the cost-effectiveness ratio.
Bernoulli Vacation Probability
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Total Expected Cost
288.366
286.954
285.543
284.131
282.719
281.308
279.896
278.485
277.073
275.662
274.250
Server Availability
0.973
0.968
0.962
0.956
0.950
0.945
0.939
0.933
0.927
0.921
0.916
CER(p)
244.845
244.843
244.841
244.839
244.837
244.834
244.832
244.830
244.827
244.825
When the server takes no vacations, the average cost-effectiveness ratio is

ACER
288.366
 296.26 dollars per additional unit of time ,
0.973
while the marginal cost-effectivess ratio when the sever always takes a vacation is

MCER
274.250  288.366
 244.82 dollars per additional unit of time .
0.916  0.973
An Unreliable General Retrial and Two Phases Queue Under Bernoulli Vacation Schedule
461
Comparing the MCER for incorporating server vacations to its ACER when there is
no server vacation suggests including server vacations rather than excluding them.
It is also possible to compute the cost-effectiveness ratio for different values of the
Bernoulli vacation schedule. Table 2 below shows the small effect of the probability of a
vacation on the cost-effectivenss ratio. However, this ratio decreases as the probability
increases.
10. Conclusion
We have studied in this paper an M/G/1 retrial queue with the following features:
each customer requires two successive phases of service, a customer who finds the server
busy joins an orbit and retries for service after a random time generally distributed, the
server is unreliable and may break down during any phase of service, and the server is
subject to a Bernoulli vacation schedule. We have obtained the following results: the
condition under which steady state is reached, the system size distribution at a departure
epoch, the probability generating function of the joint distributions of the server state and
orbit size, the decomposition property, the system availability, the failure frequency, and the
Laplace transform of the system reliability function.
This study can be complemented in various ways. For example, it may be worth
generalizing the arrival process to the case of a compound Poisson process.
Acknowledgemts
The authors would like to thank Dr. Elsayed A. Elsayed, the Editor-In-Chief of the
journal.
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Authors’ Biographies:
Gautam Choudhury holds his M.Sc and Ph.D. in Statistics from Gauhati University,
Guwahati, Assam, India. Presently, he is working as an Associate Professor in
Mathematical Sciences Division, Institute of Advanced Study in Science and Technology,
Paschim Boragoan, Guwahati, Assam, India. His areas of interest are queueing theory,
stochastic models in communication systems and statistical inference in stochastic
processes. He has numerous publications in journals of mathematics, statistics, and
operations research.
Lotfi Tadj is an Assistant Professor in the Department of Information Systems and
Decision Sciences at Fairleigh Dickinson University (Canada). He holds an MSc in
Applied Mathematics from Carnegie-Mellon University (USA) and a Ph.D. in Operations
Research from the Florida Institute of Technology (USA). His research interests include
the optimal control of queueing and inventory systems. He has numerous publications in
journals of mathematics, statistics, and operations research.
Mitali Deka holds her M.Sc. in Statistics from Gauhati University, Guwahati, Assam,
India. Presently, she is working as a Guest Faculty in Assam Institute of Management,
Guwahati, Assam. Her research interest includes stochastic processes and queueing theory.