RAIRO-Oper. Res. 48 (2014) 381–398 RAIRO Operations Research
DOI: 10.1051/ro/2014013 www.rairo-ro.org
PERFORMANCE ANALYSIS OF SINGLE SERVER
NON-MARKOVIAN RETRIAL QUEUE WITH WORKING
VACATION AND CO N STANT RETRIAL POLICY
V. Jailaxmi
1
, R. Arumuganathan
1
and M. Senthil Kumar
2
Abstract. This paper analyses an M/G/1 retrial queue with working
vacation and constant retrial policy. As soon as the system becomes
empty, the server begins a working vacation. The server works with
different service rates rather than completely stopping service during a
vacation. We construct the mathematical model and derive the steady-
state queue distribution of number of customer in the retrial group.
The effects of various performance measures are derived.
Keywords. Retrial queue, working vacation, constant retrial policy.
Mathematics Subject Classification. 60K25, 90B22.
1. Introduction
A retrial system consists of a primary service facility and an orbit. Customers
arrive at the service facility at a Poisson rate from main pool. Upon arrival of a
customer, if the server is busy or under repair or on vacation the arrival will join
the retrial group in the orbit and attempt for service again at some time later. Such
situations arise in many communication protocols, local area networks and daily
life situations. In aviation, where a plane finds the runway occupied remakes its
attempt of landing later and in this case the plane is said to be in orbit. In telephone
Received September 24, 2012. Accepted January 8, 2014.
1
Department of Mathematics, PSG College of Technology, 641004 - Coimbatore, India.
subaprasannaa@yahoo.co.in; ran
psgtech@yahoo.co.in
2
Department of Applied Mathematics and Computational Sciences, PSG College
of Technology, 641004 - Coimbatore, India. ms
kumar@yahoo.com
Article published by EDP Sciences
c
EDP Sciences, ROADEF, SMAI 2014
382 V. JAILAXMI ET AL.
where a telephone subscriber who obtains a busy signal repeats the call until the
required connection is made. The detailed overviews of the related references with
retrial queues can be found in the book of Falin and Templetion [1] and the survey
papers, of Artalejo [2, 3]. The single server retrial queue with priority calls have
been studied by Choi et al. [4–6] for many applications in telecommunication and
mobile communication.ArtalejoandGomez−Corral [7] have made a detailed study
on retrial queueing systems.
Vacation models had been the subject of interest in queueing theory in recent
years because of their applications in real life congestion situations such as man-
ufacturing and production, computer and communication systems, service and
distribution systems, etc. A comprehensive and excellent study on the vacation
models can be found in Takagi [8]. For related literature of retrial queues with
vacations, Li and Yang [9] developed an M/G/1 retrial system with server vaca-
tions and M independent identical input sources. Later Artalejo [10] analyzed an
M/G/1 retrial queue with exhausted server vacations, that is the server takes a
vacation only when there are no customers in the orbit. A literature survey on
queueing systems with server vacations can be found in Doshi [11]. Doshi [12]dis-
cussed an M/G/1 system with variable vacations. Batch arrival Markovian single
server queueing systems with multiple vacations were first studied by Baba [13].
Later Senthilkumar and Arumuganathan [14] have analyzed single server batch
arrival retrial queue with general vacation time under Bernoulli schedule and two
phases of heterogeneous service. The variations and extensions of these vacation
models can be referred to Lee et al. [15,16] and Krishna Reddy et al. [17]. Arumu-
ganathan et al. [18] analyzed a steady state non-Markovian bulk queueing system
with N-policy and different types of vacation. Haridass and Arumuganathan [19]
analyzed a batch arrival, bulk service queueing system with interrupted vacation.
A queue with working vacation was first analyzed by Servi and Finn [20], they
obtained the queue length distribution of M/M/1/Wv queue. They discussed a
classical single server vacation model in which a single server works at a different
rate rather than completely stopping during the vacation period. Further, they
applied the model for the performance evaluation of Wavelength Division Multi-
plexing (WDM) optical systems. But they have assumed exponential service time,
which may not be the case always. Subsequently, Kim et al. [21] have analyzed
an M/G/1 queue with exponentially distributed working vacations and obtained
the steady state queue length distribution through the decomposition approach.
Later Wu and Takagi [22] extended Servi and Finns model to an M/G/1 working
vacation in which, both regular service time and the service time in working vaca-
tion are assumed to be generally distributed. Li et al. [23] considered an M/G/1
queue with exponentially distributed working vacations, which is a special case of
that in Wu and Takagi [22]. All the above contributors consider classical queueing
model.
Tien Van Do [24] studied a Markovian retrial queue with working vacation. But
in practice, there must be generally distributed service times which are motivated
by the performance analysis of Media Access Control (MAC) function in wireless
A WORKING VACATION QUEUE WITH CONSTANT RETRIAL RATE 383
networks. Wireless MAC protocols often use collision avoidance techniques, in
conjunction with a (physical or virtual) carrier sense mechanism. In carrier sense
mechanism, when a node wishes to transmit a packet, it first waits until the channel
is idle. Nodes hearing RTS(Request-to-Send) or CTS(Clear-to-Send ) stay silent
for the duration of the corresponding transmission. Once channel becomes idle,
the node waits for a randomly chosen duration before attempting to transmit.
This mechanism can be modeled as M/G/1 retrial queueing model with working
vacation model by considering the orbit as pool of packets waiting for transmission
once it senses the idle channel and RTS and CTS as working vacation times. So,
in this paper we introduce an M/G/1 retrial queue with single working vacation
and constant retrial policy. Analytical treatment of this model is obtained using
supplementary variable technique. The probability generating function of number
of customers in the retrial group is obtained.
2. The mathematical model
In this paper an M/G/1 retrial queue with working vacation and constant retrial
policy is analyzed. The customers arrive according to Poisson process with rate
λ. If the server is busy at the arrival time, the customers join the orbit to repeat
their request later, whereas if the server is idle then the arriving customer begins
its service immediately. The customers in the orbit try for service one by one
with a constant retrial rate γ when the server is idle. The single server takes a
working vacation at times when the customers being served depart from the system
and no customers are in the orbit. The server works with different service rates
rather than completely stopping service during a vacation. The service rate is
μ
b
when the server is not on vacation and μ
v
during working vacation (μ
v
<
μ
b
). Vacation durations are exponentially distributed with parameter η.After
completing a vacation, the server stays idle in the system until a customer arrives
from main pool or from orbit.
Let S
v
(x)(s
v
(x)) {
˜
S
v
(θ)} [S
0
v
(x)] be the cumulative distribution function (prob-
ability density function) {Laplace transform} [remaining service time] of service
during working vacation. Let S
b
(x)(s
b
(x)) {
˜
S
b
(θ)} [S
0
b
(x)] be the cumulative dis-
tribution function (probability density function) {Laplace transform} [remaining
service time] of service when the server is not on working vacation. N(t) denotes
the number of customers in the orbit at time t. The process considered here is a
semi-Markov process which become Markov by including additional random vari-
able as the remaining service time as given by Limnios and Oprisan [25].
The server state is denoted as
C(t)=
⎡
⎢
⎢
⎢
⎢
⎣
0, if the server is idle during working vacation
1, if the server is idle and not on working vacation
2, if the server is busy during working vacation
3, if the server is busy and not on working vacation
384 V. JAILAXMI ET AL.
Now the system state probabilities are defined as follows:
1) W
n
(t)=Pr{N(t)=n, C(t)=0},n ≥ 0 is the probability that at time t the
server is idle during vacation and the orbit size n.
2) I
n
(t)=Pr{N(t)=n, C(t)=1},n ≥ 0 is the probability that at time t the
server is idle but not on working vacation and the orbit size is n.
3) Q
n
(x, t)dt = Pr{N(t)=n, C(t)=2,x ≤ S
0
v
(t) ≤ x +dt},n ≥ 0istheisthe
probability that at time t the server is busy during working vacation, the orbit
size is n and the remaining service time of a customer during working vacation
at an arbitrary time is between x and x +dt.
4) P
n
(x, t)dt = Pr{N (t)=n, C(t)=3,x ≤ S
0
b
(t) ≤ x +dt},n ≥ 0isthejoint
probability that at time t the server is busy when it is not on working vacation,
the orbit size is n and the remaining service time of a customer when the server
is not on working vacation at an arbitrary time is between x and x +dt.
3. Steady state queue size distribution
To derive the steady state queue size distribution the following equations are
obtained, using supplementary variable technique,
W
0
(t + Δt)=W
0
(t)(1 − λΔt − ηΔt)+Q
0
(0,t)Δt + P
0
(0,t)Δt
W
n
(t + Δt)=W
n
(t)(1 − λΔt − ηΔt − γΔt)+Q
n
(0,t)Δt
I
0
(t + Δt)=I
0
(t)(1 − λΔt)+W
0
(t)ηΔt
I
n
(t + Δt)=I
n
(t)(1 − λΔt − γΔt)+W
n
(t)ηΔt + P
n
(0,t)Δt
Q
n
(x − Δt, t + Δt)=Q
n
(x, t)(1 − λΔt − ηΔt)+λW
n
(t)s
v
(x)Δt
+γW
n+1
(t)s
v
(x)Δt + λQ
n−1
(x, t)Δt(1 − δ
n,0
)
P
n
(x − Δt, t + Δt)=P
n
(x, t)(1 − λΔt)+λI
n
(t)s
b
(x)Δt + γI
n+1
s
b
(x)Δt
+
∞
0
Q
n
(y,t)dy
ηs
b
(x)Δt + λP
n−1
(x, t)(1 − δ
n,0
)Δt
where δ
n,0
=
0 if n =0
1 if n =0
.
In steady state, we can set W
0
= lim
t→∞
W
0
(t), I
0
= lim
t→∞
I
0
(t), W
n
=
lim
t→∞
W
n
(t), I
n
= lim
t→∞
I
n
(t) and limiting densities Q
n
(x) = lim
t→∞
Q
n
(x, t)
for x>0andP
n
(x) = lim
t→∞
P
n
(x, t)forx>0.
Now the above equations under steady state conditions can be written as follows:
(λ + η)W
0
= Q
0
(0) + P
0
(0) (3.1)
(λ + γ + η)W
n
(0) = Q
n
(0) (3.2)
λI
0
= ηW
0
(3.3)
(λ + γ)I
n
(0) = W
n
(0)η + P
n
(0) (3.4)
A WORKING VACATION QUEUE WITH CONSTANT RETRIAL RATE 385
−d
dx
Q
n
(x)=− (λ + η)Q
n
(x)+λW
n
(0)s
v
(x)
+ γW
n+1
(0)s
v
(x)+λQ
n−1
(x)(1 − δ
n,0
) (3.5)
−d
dx
P
n
(x)=− λP
n
(x)+λI
n
(0)s
b
(x)+γI
n+1
(0)s
b
(x)
+ λP
n−1
(x)(1 − δ
n,0
)+
∞
0
Q
n
(y)dy
ηs
b
(x). (3.6)
Assume that
Laplace transform(P
n
(x)) =
˜
P
n
(θ)=
∞
0
e
−θx
P
n
(x)dx;
Laplace transform(Q
n
(x)) =
˜
Q
n
(θ)=
∞
0
e
−θx
Q
n
(x)dx.
Taking Laplace transform on steady state equations (5) and (6), we have
θ
˜
Q
n
(θ) − Q
n
(0) = (λ + η)
˜
Q
n
(θ) − λW
n
(0)
˜
S
v
(θ) − γW
n+1
(0)
×
˜
S
v
(θ) − λ
˜
Q
n−1
(θ)(1 − δ
n,0
) (3.7)
θ
˜
P
n
(θ) − P
n
(0) = λ
˜
P
n
(θ) − λI
n
(0)
˜
S
b
(θ)
− γI
n+1
(0)
˜
S
b
(θ) − λ
˜
P
n−1
(θ)(1 − δ
n,0
) −
˜
Q
n
(0)η
˜
S
b
(θ). (3.8)
The following generating functions are helpful in deriving the probability generat-
ing function of orbit size.
W (z, 0) =
∞
n=0
W
n
(0)z
n
; I(z,0) =
∞
n=0
I
n
(0)z
n
;
˜
Q(z,θ)=
∞
n=0
˜
Q
n
(θ)z
n
; Q(z, 0) =
∞
n=0
Q
n
(0)z
n
(3.9)
˜
P (z, θ)=
∞
n=0
˜
P
n
(θ)z
n
; P (z,0) =
∞
n=0
P
n
(0)z
n
where |z|≤1.
Multiplying equations (1) and (3) by z
0
, equations (2), (4), (7) and (8) by z
n
,
taking summation from n =0to∞ and using (9), we get,
(λ + η)W (z, 0) + γ(W (z,0) − W
0
)=Q(z,0) + P
0
(3.10)
λI(z,0) + γ(I(z, 0) − I
0
)=ηW(z,0) + (P (z,0) − P
0
) (3.11)
(θ − (λ + η)+λz)
˜
Q(z,θ)=Q(z,0) − λW (z,0)
˜
S
v
(θ) −
γ
z
(W (z,0) − W
0
)
˜
S
v
(θ)
(3.12)
(θ − λ + λz)
˜
P (z, θ)=P (z,0) −
λ +
γ
z
˜
S
b
(θ)I(z,0) +
γ
z
×
˜
S
b
(θ)I
0
−
˜
Q(z,0)η
˜
S
b
(θ). (3.13)
