The object of this paper is to study joint and marginal distributions of the system states at any arbitrary time moment for a single server, finite source retrial queue, in which the server can sustain breakdowns during service times. The server life times as well as the intervals between repetitions are exponentially distributed, while the repair and the service times are generally distributed. Unlike the unreliable model studied by J. Wang et al. [23], in which the interrupted customer waits for the server back from repair, to accomplish the remaining service, in our model this customer goes to the orbit, losing the service time, elapsed before the breakdown occurs

Dragieva, V.

English

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SYSTEM STATE DISTRIBUTIONS
IN ONE FINITE SOURCE
UNRELIABLE RETRIAL QUEUE
V. Dragieva
University of Forestry
Sofia, Bulgaria
vildrag2001@yahoo.com
The object of this paper is to study joint and marginal distributions of the system states
at any arbitrary time moment for a single server, finite source retrial queue, in which the
server can sustain breakdowns during service times. The server life times as well as the
intervals between repetitions are exponentially distributed, while the repair and the service
times are generally distributed. Unlike the unreliable model studied by J. Wang et al. [23],
in which the interrupted customer waits for the server back from repair, to accomplish the
remaining service, in our model this customer goes to the orbit, losing the service time,
elapsed before the breakdown occurs.
Keywords: closed queueing system, retrial queue, unreliable server.
1. INTRODUCTION
Retrial queues are widely used to model many real situations in telephone, computer,
communication and other systems. The most obvious example of a retrial queue appears in a
telephone call, when a person phones and finds the line engaged. Usually in such a situation
the subscriber repeats his attempts until he is satisfied. A description of other situations in
which retrials arise can be found in [11], and more recent applications of retrial models
in [6].
There is a large literature devoted to retrial queues in the past twenty years, but most
papers assume that the population of potential customers is very large so that input stream is
Poisson. However, in many practical situations, the number of users who access the system
is finite. Thus, it is important to take into account the fact that the rate of generation of
new calls changes, depending of the number of customers in the system. This can be done
with the help of finite - source or quasi - random input models. Finite source retrial models
arise in various practical areas as magnetic disk memory systems, [18], a star-like local area
networks, [12] and [17], a local area networks with nonpersistent CSMA/CD protocol, [16],
telephone networks, [14], etc. A systematic account of the fundamental methods and results
on this topic can be found in [4–6, 11] and the literature therein.
Another very important component of queueing models is the reliability of the server/ser-
vers. More and more applications of these models in telephone, banking and airline companies
depend on the correct and timely operation of the system and require a particular quality of
service. Although the reliability study is of great importance, there are only few works that
10
take into consideration retrial phenomenon involving the reliability of the server/servers.
Most of them deal with unreliable retrial queues with an infinite customers source [1, 3, 7,
15, 20–22, 24]. Finite source retrial systems with unreliable servers are still an interesting
topic. In fact, in the retrial queueing literature we found only a few papers by Sztrik et
al. [2, 19] where various types of unreliable systems are studied by using the software
MOSEL (Modeling, Specification and Evaluation Language) and a paper of Garbi [13], that
studies unreliable retrial systems with several homogeneous servers by means of GSPNs
(Generalized Stochastic Petri Nets). To the best of our knowledge the only paper, providing
an exhaustive theoretical analysis of a single server finite source retrial queue with unreliable
server is the paper [23]. With the help of supplementary variables method combined with
discrete transformations approach they investigate the stationary characteristics of the model,
including the queue length process, busy period and waiting time process.
In the model, considered by Wang et al., [23] it is assumed that the server sustains
breakdowns only when it is busy and immediately after the breakdown is sent for repair.
The customer, being served just before the breakdown occurs waits for the server back to
accomplish its remained service. In practice there are many other opportunities regarding
the breakdown occurrences, the behavior of the server and of the customer, in case of server
breakdown.
In present paper we consider a single server unreliable finite source retrial model in
which, like in the model studied in [23] breakdowns occur only when the server is busy
and after the breakdown the server is immediately sent for repair. But in our model, unlike
the model in [23] this customer is sent back to the orbit and repeats his attempts for a new
service. The service time, spent before the breakdown is just “lost”. For convenience further
in the paper we shall call the model studied in [23] “type 1 unreliable model/system” and
the model, considered in present paper - “type 2 unreliable model/system”.
The main objective of the paper is to establish close/recurrent formulas for computing
stationary distributions of the orbit size and the server state and their main characteristics.
It is done using the supplementary variables method as well as the discrete transformation
method.
2. MODEL DESCRIPTION AND NOTATION
We consider a retrial queueing system of type M/G/1//N in Kendall’s notation with
server breakdowns. This means that the system has one server, which serves N customers
(subscribers), 2 ≤ N < ∞ and each one of them, in his free state produces a Poisson
process of demands (calls) with rate λ. These customers are identified as primary customers
or sources of primary calls. Thus, when a source is free at time moment t (i.e. is not being
served and is not waiting for service) it may generate a primary call during interval (t, t+dt)
with probability λdt. If the server is free at the instant of a primary call arrival, then the call
starts to be served. During the service time the source cannot generate a new primary call.
After service the source moves into the free state and can generate a new primary call.
When an arriving primary customer finds the server busy, he leaves the service area
and repeats the request after some random time. Between trials a customer is said to be in
11
orbit (queue) or to be a source of secondary calls, a secondary subscriber. Each secondary
subscriber generates a Poisson flow of repeated calls with intensity µ until he finds the
server free. As in the case of a primary call, after service the source becomes free and can
generate a new primary call.
The server has an exponentially distributed lifetime with failure rate α during service
processes. When the server breaks down it is sent immediately for repair and the customer
being served just before server breakdown goes to the orbit, “losing” the service time, spent
before the breakdown. The behavior of the interrupted customer differs our model (as it is
called in the introduction “type 2 unreliable model”) from the unreliable queueing model
studied in [23], “type 1 unreliable model”, where the interrupted customer waits for the
server back to accomplish his remaining service.
We assume that primary calls, repeated attempts, service times, breakdowns and repairs
are mutually independent. The service times are identically distributed both for the primary
and secondary subscribers with probability distribution function G(x), G(0) = 0, hazard
rate function γ(x),
γ(x) =
G′(x)
1−G(x)
,
Laplace - Stieltjes transform g(s) and expected value ν−1. The server repair times follow
an arbitrary distribution with distribution function, hazard rate function, Laplace - Stieltjes
transform and expected value B(x), β(x), b(s) and κ−1, respectively.
The state of the system at time t can be described by the process {X(t), t ≥ 0} =
{C(t), R(t), z(t), w(t), t ≥ 0}, where C(t) describes the server state,
C(t) =


0, if the server is free,
1, if the server is busy,
2, if the server is down (under repair),
R(t) is the number of sources of repeated calls at time t (secondary subscribers, orbit size,
queue length), z(t) and w(t) are supplementary variables, equal to the elapsed service time
(when C(t) = 1) and elapsed repair time (when C(t) = 2), respectively. It is obvious
that {X(t), t ≥ 0} is a Markov process and since the situation C(t) = 0, R(t) = N is
impossible, its state space is the set {0, 1, 2} × {0, 1, . . . , N − 1}.
We define the following probabilities (densities):
p1n(x)dx = lim
t→∞
P {C(t) = 1, R(t) = n, x ≤ z(t) < x+ dx} , (1)
p2n(x)dx = lim
t→∞
P {C(t) = 2, R(t) = n, x ≤ w(t) < x+ dx} , (2)
pin = lim
t→∞
P {C(t) = i, R(t) = n} , (3)
i = 0, 1, 2, n = 0, 1, . . . , N − 1 and obviously
p1n =
∫
∞
0
p1n(x)dx, p2n =
∫
∞
0
p2n(x)dx.
12
3. DISTRIBUTIONS OF THE ORBIT SIZE AND THE
SERVER STATE AT ANY ARBITRARY TIME MOMENT
In this section we derive formulas for computing joint distributions pin(x), i = 1, 2, p0n
of the orbit size and the server state at any arbitrary time moment, so called an outside
observer’s distribution. To this end we obtain in a general way the equations of statistical
equilibrium,
[(N − n)λ+ nµ]p0n =
∫
∞
0
p1n(x)γ(x)dx+
∫
∞
0
p2n(x)β(x)dx, (4)
dp1n(x)
dx
= −[(N − n− 1)λ+ α + γ(x)]p1n(x)
+ (N − n)λp1,n−1(x), (5)
dp2n(x)
dx
= −[(N − n)λ+ β(x)]p2n(x)
+ (N − n+ 1)λp2,n−1(x), (6)
with boundary conditions
p1n(0) = (N − n)λp0n + (n+ 1)µp0,n+1, (7)
p2n(0) = α
∫
∞
0
p1,n−1(x)dx = αp1,n−1, (8)
n = 0, 1, . . . , N − 1, p1,−1(x) = p2,−1(y) = p0,N = 0.
As in most cases of finite queues investigation (see [9,10,23]) we will solve these equations
with the help of so-called “discrete transformations” approach. Namely, to simplify the
differential equations (5) we introduce variables q1m(x) and qim, “discrete transformations”
of p1m(x) and pim, i = 0, 1, related with them by the equations
p1n(x) =
n∑
m=0
(−1)m
(
N − n− 1 +m
m
)
q1,N−n−1+m(x), (9)
pin =
n∑
m=0
(−1)m
(
N − n− 1 +m
m
)
qi,N−n−1+m, i = 0, 1. (10)
Unfortunately, unlike the reliable model, studied in [10] and the type 1 unreliable model,
studied in [23], these transformations cannot simplify to the desired form all equations
of statistical equilibrium, especially the differential equations (6). To simplify them we
introduce variables q2m(x) and q2m, “discrete transformations” of p2m(x) and p2m, defined
by the formulas
p2n(x) =
n∑
m=0
(−1)m
(
N − n+m
m
)
q2,N−n−1+m(x), (11)
p2n =
n∑
m=0
(−1)m
(
N − n+m
m
)
q2,N−n−1+m. (12)
13
Proposition. The quantities q1m(x) and q2m(x) are solutions of the following systems
differential equations:
dq1m(x)
dx
= −[mλ+ α + γ(x)]q1m(x), (13)
dq2m(x)
dx
= −[(m+ 1)λ+ β(x)]q2m(x), (14)
with boundary conditions
q1m(0) = [(m+ 1)λ+ (N − 2m− 1)µ] q0m
+ (N −m)µq0,m−1 + (m+ 1)(λ− µ)q0,m+1, (15)
q2m(0) = α
∫
∞
0
q1,m−1(x)dx = αq1,m−1, (16)
m = 0, 1, . . . , N − 1.
Equations, relating quantities q0m with q1m(x) and q2m(x) as well as the normalizing
condition are respectively:
[(m+ 1)λ+ (N −m− 1)µ]q0m + (m+ 1)(λ− µ)q0,m+1
=
∫
∞
0
q1m(x)γ(x)dx+
N−1∑
k=m
(−1)k−m
∫
∞
0
q2k(x)β(x)dx, (17)
q00 + q10 +
N−1∑
m=0
(−1)mq2m = 1. (18)
Here
q0,−1 = q0N = q1,−1 = q1,−1(x) = 0.
Solving the system (13) - (18) we obtain formulas for computing quantities qim(x),
m = 1, 2 and qim, i = 0, 1, 2.
Corollary 1. Quantities qim(x), i = 1, 2 and qim, i = 0, 1, 2 can be calculated by the
formulas
q1m(x) = q1m(0)e
−(mλ+α)x(1−G(x)), (19)
q2m(x) = q2m(0)e
−(m+1)λx(1− B(x)), (20)
for q2m(0) it holds
q2m(0) = αq1,m−1(0)
1− g((m− 1)λ+ α)
(m− 1)λ+ α
, (21)
while q1m(0) and q0m are related by the linear equations
[(m+ 1)λ+ (N −m− 1)µ]q0m + (m+ 1)(λ− µ)q0,m+1 = q1m(0)g(mλ+ α)
14
+ α
N−1∑
k=m
(−1)k−mq1,k−1(0)
1− g((k − 1)λ+ α)
(k − 1)λ+ α
b((k + 1)λ) (22)
and equations (15), 0 ≤ m ≤ N − 1. In addition, q1m(0) and q0m, 0 ≤ m ≤ N − 1 have to
satisfy the following normalizing condition:
q00 + q10(0)(1− g(α))
(
1
α
−
b(2λ)
2λ
)
− α
N−2∑
k=1
(−1)kq1k(0)
1− g(kλ+ α)
kλ+ α
1− b((k + 2)λ)
(k + 2)λ
= 1. (23)
All of the quantities qim(x), i = 1, 2 and qim, i = 0, 1, 2 for which m < 0 or m > N − 1 are
equal to 0.
Thus, the computational difficulty for calculating each of the quantities qim(x),m = 1, 2
and qim, i = 0, 1, 2 is to solve the linear equations (15), (22), (23). To solve them we can,
for example, eliminate from (15), (22) q1m(0) and get equations of the type:
N−1∑
k=m−2
Qkmq0k = 0, 1 ≤ m ≤ N − 1, Q1,−1 = 0. (24)
From here we express all q0k, k = 0, ..., N − 3 in terms of q0,N−1 and q0,N−2 and then find
q0,N−1 and q0,N−2 from the normalizing condition (23)) and equation (24) for m = 1.
Corollary 2. With the help of quantities q0m and/or q1m(0) we can calculate many of the
basic system characteristics, without calculating the probabilities pim, i = 0, 1, 2 :
• The stationary probability that the server is idle, P0,
P0 =
N−1∑
n=0
p0n = q00;
• The stationary probability that the server is working, P1,
P1 =
N−1∑
n=0
p1n = q10 = q10(0)
1− g(α)
α
;
• The stationary probability that the server is under repair, P2,
P2 =
N−1∑
n=0
p2n =
N−1∑
m=0
(−1)mq2m
=
N−2∑
l=0
(−1)l+1αq1l(0)
1− g(lλ+ α)
lλ+ α
1− b((l + 2)λ)
(l + 2)λ
;
• The stationary mean orbit size, E[R],
E[R] =
N−1∑
n=0
n (p0n + p1n + p2n)
= N − (q00 + q10 + q20 + q01 + q11) . (25)
15
4. CONCLUSION
In present paper we establish formulas for computing the stationary distributions of
the orbit size and server state at any arbitrary time moment and some of their main
characteristics in one finite source, unreliable retrial queue. These formulas are the basis for
any further study of the model, like system state distributions at the moments of a primary
call arrivals, waiting time process and other. They also allow numerical studies of the system
characteristics dependence on the system parameters to be carried out.
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System State Distributions In One Finite Source Unreliable Retrial Queue

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System State Distributions In One Finite Source Unreliable Retrial Queue

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