In this work, we carry out a stochastic analysis of the M/G/1 retrial queue with batch arrivals and impatient customers. Our study includes: steady state joint distribution of the server state and the number of customers in retrial group, embedded Markov chain and stochastic decomposition for the number of customers in the system. We investigate also the asymptotic behaviour of the system under high and low retrial intensities.0.10.136

Arrar, K. N.

Baillon, J. B.

Djellab, N. V.

English

Электронная библиотека БГУ

ON THE M
X
=G=1 RETRIAL QUEUE
WITH IMPATIENT CUSTOMERS
N.V. Djellab
1
, K.N. Arrar
1
, J.B. Baillon
2
1
University of Annaba
2
University Paris 1 Pantheon Sorbone
1
Annaba, Algeria
2
UFR 27, CERMSEM CNRS-UMR 8095, Frane
djellabyahoo.fr
In this work, we arry out a stohasti analysis of the M=G=1 retrial queue with
bath arrivals and impatient ustomers. Our study inludes: steady state joint dis-
tribution of the server state and the number of ustomers in retrial group, embedded
Markov hain and stohasti deomposition for the number of ustomers in the system.
We investigate also the asymptoti behaviour of the system under high and low retrial
intensities.
Keywords: retrial queue, bath arrival, impatient ustomer, deomposition property,
low retrial rate, onvergene.
1. INTRODUCTION
In telephone networks, we an observe that a alling subsriber after some unsuess-
ful retrials gives up further repetitions and abandons the system. In queueing systems
with repeated attempts (retrial queues), this phenomenon is represented by the set of
probabilities fH
k
; k  1g, alled the persistene funtion, where H
k
is the probability
that after the kth attempt fails, a ustomer will make the (k + 1)th one. In general,
it is assumed that the probability of a ustomer reinitiating after failure of a repeated
attempt does not depend on the number of previous attempts (i.e. H
2
= H
3
= :::) [2℄.
In this paper, we onsider a single server queueing system at whih primary ustomers
arrive in bathes of size k (with probability 
k
(t); k  1) aording to a Poisson stream
with rate  > 0. If the server is busy at the arrival epoh, then with probability 1 H
1
all these ustomers leave the system without servie and with probability 0 < H
1
< 1
join the retrial group (orbit); whereas if the server is idle, then one of the arriving us-
tomers begins his servie and leaves the system after the servie and the others go to
the orbit. Let 
k
= lim
k!1

k
(t); k  1. Thus, C(z) =
1
P
k=1

k
z
k
is the generating funtion
of the steady state distribution of the bath size and  = C
0
(1) is the mean bath size.
The impatiene phenomenon is represented by the probability 1 H
1
(the probability
H
2
= 1). Any orbiting ustomer will repeatedly retry until the time at whih he nds
the server idle and starts his servie. The retrial times are exponentially distributed
with distribution funtion T (x) = 1   e
 x
; x  0, having nite mean
1

. The servie
49
times follow a general distribution with distribution funtion B(x) and Laplae-Stieltjes
transform
e
B(s); Re(s) > 0. Let 
k
= ( 1)
k
e
B
(k)
(0) be the kth moment of the servie
time about the origin and  = H
1

1
 be the tra intensity. Finally, we admit the
hypothesis of mutual independene between all random variables dened above.
2. STEADY STATE DISTRIBUTION OF THE SYSTEM
STATE AND DECOMPOSITION PROPERTY
The state of the onsidered system at time t an be desribed by means of the
proess fC(t); N
o
(t); (t); t  0g, where N
o
(t) is the number of ustomers in the orbit,
C(t) is the state of the server at time t. We have that C(t) is 0 or 1 depending on
whether the server is idle or busy. If C(t) = 1, (t) represents the elapsed servie time
of the ustomer in servie at time t.
Let  < 1. Dene
p
0n
= lim
t!1
P (C(t) = 0; N
o
(t) = n);
p
1n
=
1
Z
0

lim
t!1
d
dx
P (C(t) = 1; (t)  x;N
o
(t) = n)

dx:
In the previous work [1℄, we found the partial generating funtions
P
0
(z) =
1
X
n=0
z
n
p
0n
=
H
1
(1  )
 +H
1
(1  )
e
[


z
R
1
1 
e
B(H
1
(1 C(u)))
C(u)
u
e
B(H
1
(1 C(u))) u
du
#
;
P
1
(z) =
1
X
n=0
z
n
p
0n
=
1 
e
B(H
1
(1  C(z)))
(
e
B(H
1
(1  C(z)))  z)H
1
P
0
(z):
We also investigated the steady state queue size distribution at departure epohs 
k
(the
time when the server enters the idle state for the k-th time). The obtained generating
funtion of the steady state distribution of the embedded Markov hain has the following
form:
'(z) =
1  

e
B(H
1
(1  C(z)))(1  C(z))
e
B(H
1
(1  C(z)))  z
e
[


z
R
1
1 
e
B(H
1
(1 C(u)))
C(u)
u
e
B(H
1
(1 C(u))) u
du
#
:(1)
At present, we establish that the right hand part of (1) an be deomposed into two
fators: the rst fator
1 

e
B(H
1
(1 C(z)))(1 C(z))
e
B(H
1
(1 C(z))) z
is the generating funtion for the num-
ber of ustomers at departure epohs in the ordinary M
X
=G=1 queue with impatient
ustomers (model M
1
), and the seond one e
[


z
R
1
1 
e
B(H
1
(1 C(u)))
C(u)
u
e
B(H
1
(1 C(u))) u
du
#
=
P
0
(z)
P
0
(1)
represents
the generating funtion for the number of ustomers at the departure epohs in the
orresponding retrial queue (model M

) given that the server is idle. Thus, stohas-
ti deomposition property of the onsidered system an be expressed in the following
manner:
f0; N
o
(t); t  0g = f0; N
q1
(t); t  0g+ f0; R

(t); t  0g:
50
The proesses f0; N
o
(t); t  0g and f0; R

(t); t  0g are related to the model M

,
where R

(t) represents the number of ustomers in the orbit at time t given that the
server is idle. The proess f0; N
q1
(t); t  0g is assoiated with the model M
1
, where
N
q1
(t) is the number of ustomers in the waiting line at time t.
3. ASYMPTOTIC BEHAVIOUR OF THE SYSTEM
Now, we investigate the asymptoti behaviour of the system under low retrial rate
( ! 0). To this end, the following result is established:
Theorem 1. If 
2
< 1, then as  ! 0, the number of ustomers in the orbit is
asymptotially Gaussian with mean


+ 1
1 
and variane



1
+
2
1
+C
00
(1)
2
1
 
1

3
 
2

2
1
+
2

2
2(1 )
2

2
1
.
In general, under high retrial intensity ( ! 1) the steady state distribution of a
retrial queue onverges to a limit, whih is usually the steady state distribution of a
ertain limit system. In our ase, it is intuitive that this is the model M
1
. We prove
this heuristi argument with the help of the stohasti deomposition property.
Let

n
() = lim
k!1
P (N
o
(
k
) = n);

n
(1) = lim
k!1
P (N
q1
(
k
) = n);
q
n
() = lim
k!1
P (R

(
k
) = n):
Theorem 2. The following inequalities take plae
2
1  

e
B(H
1
)
(1  q
0
()) 
1
X
n=0
j
n
()  
n
(1)j  2(1  q
0
()); where
q
0
() = e
[


z
R
1
1 
e
B(H
1
(1 C(u)))
C(u)
u
e
B(H
1
(1 C(u))) u
du
#
and =H
1

1
:
As  !1;
1
X
n=0
j
n
()  
n
(1)j = 0

1


:
REFERENCES
1. Djellab N., Arrar K. N., Baillon J. B. Asymptoti behaviour of the number of
ustomers in retrial group of M/G/1 retrial queue with bath arrivals // Interna-
tional Conferene Modern Stohastis: Theory and Appliations II, Kyev, Ukraine.
2010. P. 711.
2. Falin G. I., Templeton J. G. C. Retrial Queues // Chapman and Hall 1997.
51


On The Mx/G/1 Retrial Queue With Impatient Customers

http://elib.bsu.by/bitstream/123456789/37489/1/Djellab_Arrar_Baillon.pdf

On The Mx/G/1 Retrial Queue With Impatient Customers

Abstract

Similar works

Full text

Available Versions

Электронная библиотека БГУ