APPROXIMATE SOLUTIONS FOR SOME ADVANCED
MULTISERVER RETRIAL QUEUES
Yang Woo Shin
1
and Dug Hee Moon
2
1
Department of Statistics, Changwon National University, 641-773, Changwon, Republic of Korea
2
Department of Industrial and Systems Engineering, Changwon National University,
641-773, Changwon, Republic of Korea
Keywords:
Multiserver retrial queue, PH-retrial time, Impatient customers, Multiclass customers, Approximations.
Abstract:
Retrial queues have been widely used for modelling many practical problems arising in computer and commu-
nication systems. It has been known to be difficult problems to develop a numerical algorithm or an approxi-
mate solution for advanced multiserver retrial queues such as the models with phase type distribution of retrial
time, impatient customers governed by a general persistence function and multiclass of customers. Recently,
we have developed an approximation method based on the approach in Fredericks and Reisner (1979) with
some modifications for the advanced systems described above. In this paper, we introduce the approximation
results developed recently.
1 INTRODUCTION
The retrial queueing system consists of a service facil-
ity with finite capacity and an orbit of an infinite size.
A customer enters the service facility if the service
facility is not full upon arrival. Otherwise, the cus-
tomer joins orbit and repeats its request after random
amount of time. The time interval between two con-
secutive attempts of each customer in orbit is called a
retrial time.
Retrial queue has been widely used for modelling
many practical problems arising in computer and
communication systems. Even for the basic M/M/c
retrial queues with multiple servers, the exact results
have not been obtained except for some special cases.
Instead, attempts to develop algorithmic or approxi-
mation methods have been made for long time. For
the Markovian retrial queues with multiple servers
with Poisson arrival, exponential service and expo-
nential retrial time, some algorithms and approxi-
mations are presented, e.g. (Neuts and Rao, 1990;
Greenberg and Wolf, 1987; Fredericks and Reisner,
1979) and for more details of retrial queues, refer
the monographs (Artalejo and G
´
omez-Corral, 2008;
Falin and Templeton, 1997) and references therein.
Neuts and Rao (1990) propose a generalized trunca-
tion method that uses the system in which only finite
number of customers in orbit can retry for approxima-
tion of the basic M/M/c retrial queue. Neuts and
Rao’s method uses the matrix analytic method for
computing the approximate system and its feasibil-
ity depends on the size of matrix components of the
approximate system. Greenberg and Wolf (1987)
present an approximation for the stationary distribu-
tion of the number of customers in service facility in
the M/M/c/K retrial queue under the assumption that
retrials see time averages (RTA). The approximation
using RTA assumption does not depend on the retrial
rate and works well only for small value of retrial rate.
Fredericks and Reisner (1979) develop an approxima-
tion for the number of customers at service facility in
M/M/c retrial queue with impatient customers. Fred-
ericks and Reisner’s method reflects the retrial rate
and works well when the retrial rate γ is small, but it
becomes worse as γ increases (Artalejo and G
´
omez-
Corral, 2008).
Kulkarni and Liang (1997) suggest some open
problems which include developments of analytical,
numerical, and approximate solutions for advanced
models such as the retrial queue with general retrial
time, impatient customers, general service time. Re-
cently, we have developed an approximation method
based on the approach in Fredericks and Reisner
(1979) with some modifications for the advanced sys-
tems. Objective of this paper is to introduce the
recent results developed by Shin and Moon (Shin
and Moon, 2011a; Shin and Moon, 2011b; Shin and
Moon, 2011c).
195
Woo Shin Y. and Hee Moon D..
APPROXIMATE SOLUTIONS FOR SOME ADVANCED MULTISERVER RETRIAL QUEUES.
DOI: 10.5220/0003716101950199
In Proceedings of the 1st International Conference on Operations Research and Enterprise Systems (ICORES-2012), pages 195-199
ISBN: 978-989-8425-97-3
Copyright
c
2012 SCITEPRESS (Science and Technology Publications, Lda.)
Fredericks and Reisner’s method with modifica-
tion is described briefly in Section 2. Approximations
of the M/M/c retrial queue with phase type retrial
time, impatient customers and multiclass customers
are briefly described in Sections 3-5.
2 APPROXIMATION OF THE
BASIC MODEL
Consider an M/M/c retrial queue with no waiting
space in service facility. Customers arrive from out-
side according to a Poisson process with rare λ and the
service time and inter retrial time distributions are ex-
ponential with rates µ and γ, respectively. Let C(t) be
the number of customers at service facility and N(t)
be the number of customers in orbit at time t in the
M/M/c retrial queue. Assume the stability condition
ρ = λ/() < 1. Then X
X
X = {(C(t)(t),N(t))t 0} is a
Markov chain on the state space S = {0, 1,2··· ,c} ×
{0,1,2,···}.
Let (C, N) be the stationary version of (C(t),N(t))
and P(i, j) = P(C = i,N = j), (i, j) S . The balance
equations are given as follows:
(λ + + jγ)P(i, j) = λP(i 1, j)
+ (i + 1)µP(i + 1, j) + ( j + 1)γP(i 1, j + 1),
0 i < c, j 0 (1)
(λ + )P(c, j) = λP(c 1, j) + λP(c, j 1)
+ ( j + 1)γP(c 1, j + 1), j 0, (2)
where P(i, j) = 0 for (i, j) 6∈ S . Define
P
i
= P(C = i), N
i
= E[N|C = i], 0 i c.
Summing over j in (1) yields
(λ
i
+ µ
i
)P
i
= λ
i1
P
i1
+ µ
i+1
P
i+1
, 0 i < c, (3)
where µ
i
= , P
1
= 0 and
λ
i
= λ + γN
i
, 0 i c 1. (4)
Thus the stationary distribution of the number of
busy servers is identical to that of finite birth-and-
death process with birth rates {λ
j
}
c1
j=0
and death rates
{µ
j
}
c
j=1
and is given by
P
i
= P
0
i
k=1
λ
k1
µ
k
, i = 1, 2,· ·· ,c (5)
with
P
0
=
"
1 +
c
i=1
i
k=1
λ
k1
µ
k
#
1
.
Let u
j
= P(N = j), j 0. Summing over i in (1) and
(2), we have that
( j + 1)γu
j+1
jγu
j
= λ(P(c, j) P(c, j 1))
+ γ(( j + 1)P(c, j + 1) jP(c, j)), j 0, (6)
and hence the mean L = E[N] is given by
L =
λ
c
P
c
γ
, (7)
where λ
c
= λ + γN
c
. Let R
i
be the proportion of re-
turning customers from orbit who find the service fa-
cility in state i, that is,
R
i
=
γN
i
P
i
c
j=0
γN
i
P
i
=
N
i
P
i
L
, 0 i c. (8)
It can be seen from (7) and (8) that
λ
c
=
λ
1 γR
c
.
We have from (4) and (8) that
λ
i
= λ +
λ
i
·· ·λ
c
µ
i+1
·· ·µ
c
R
i
, 0 i c 1. (9)
From the observations above, we adopt the follow-
ing approximation assumption.
Assumption A. The service facility behaves like a
birth-and-death process with birth rates {λ
i
}
c1
i=0
and
death rates {µ
i
}
c
i=1
and is independent of the retrials.
Let Q be the generator of the birth-and-death pro-
cess with birth rates {λ
i
}
c1
i=0
and death rates {µ
i
}
c
i=1
and q
i j
(t) be the transition probability of Q and ˜q
i j
(θ)
the Lapalce transform of q
i j
(t). To determine R
i
, sup-
pose that at time 0, a customer attempts to get service
and it finds that all the servers are busy and joins orbit.
This customer returns after an exponential time with
parameter γ. Then, R
i
is approximated by the fraction
of returning customers from orbit that find the service
facility in state i, that is,
R
i
Z
0
q
ci
(t)γe
γt
dt = γ ˜q
ci
(γ), 0 i c. (10)
Remark. The approximation of R
i
in (10) is the
same as that in Fredericks and Reisner (1979).
Once an approximation of λ
λ
λ is given, the square
matrix
˜
Q(θ) = ( ˜q
i j
(θ)) can be calculated by the for-
mula
˜
Q(θ) = (θI Q)
1
,
where I is the identity matrix of order c + 1.
Combining (9) and (10), we have the approxima-
tion formula for λ
i
as follows
λ
i
= λ+λ
λ
i
·· ·λ
c1
µ
i+1
·· ·µ
c
γ ˜q
ci
(γ)
1 γ ˜q
cc
(γ)
, 0 i c 1, (11)
ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems
196
Now we describe the computational procedure for λ
j
.
Let λ
λ
λ = (λ
0
,··· ,λ
s1
) and denote by g(λ
λ
λ) the right
hand side of the equation (11) and define a sequence
of λ
λ
λ
(n)
= (λ
(n)
0
,··· ,λ
(n)
s1
) by
λ
λ
λ
(n)
= g(λ
λ
λ
(n1)
), n = 1, 2,· ·· (12)
with λ
λ
λ
(0)
= (λ,··· , λ). Repeating the successive sub-
stitution (12), we choose λ
λ
λ
(n)
as an approximation of
λ
λ
λ if it satisfies ||λ
λ
λ
(n)
λ
λ
λ
(n1)
||
< ε for given toler-
ance ε > 0.
Numerical experiments provide that approxima-
tions for L
0
is the same as the exact result L
0
=
λ
µ
.
For improving the accuracy of approximation for L =
E[N], a modification of approximation for L is pro-
vided as follows
ˆ
L(m) = L
Appr
(m) + (L
M/M/c
L
Appr
(m
)), (13)
where L
Appr
(m) is the approximation for L in the
system with mean retrial time m and L
M/M/c
is the
mean number of customers in queue for the ordinary
M/M/c queue and m
is chosen to be large enough so
that the variation of L
Appr
(m) for m m
is negligible.
3 RETRIAL QUEUE WITH
PH-RETRIAL TIME
Consider an M/M/c retrial queue in which the re-
trial time is of phase type distribution PH(α
α
α,Γ
Γ
Γ). For
stability of the system ρ =
λ
< 1 is assumed. Shin
and Moon (2011a) proposed an approximation for this
system as follows.
Let m be the number of phases of PH(α
α
α,Γ
Γ
Γ). Let
X
0
be the number of customers at service facility
and X
i
the number of customers in orbit whose re-
trial phase is of i in stationary state and L
i
= E[X
i
],
0 i m. Then, it can be shown that π
j
= P(X
0
= j),
0 j c satisfies the balance equation of the a birth-
and-death process with birth rates {λ
j
}
c1
j=0
and death
rates {µ
j
}
c
j=1
, where µ
j
= jµ and λ
j
can be expressed
in terms of R
j
that is the proportion of returning cus-
tomers from orbit who find the service facility in state
j as
λ
j
= λ + λ
c
i= j+1
λ
i1
µ
i
!
R
j
1 R
c
, 0 j < c.
Under the approximation assumption that the service
facility behaves like a birth-and-death process with
birth rates {λ
j
}
c1
j=0
and death rates {µ
j
}
c
j=1
and is
independent of the retrials, R
j
is approximated as the
probability that a returning customer finds the service
facility in state j, that is,
R
j
Z
0
q
c j
(t)α
α
αexp(Γ
Γ
Γt)Γ
Γ
Γ
0
dt, 0 j c,
where q
i j
(t) is the transition probability of Q.
Then, λ
j
is calculated by successive substitution.
Numerical experiments provide that approximations
for L
0
is the same as the exact result L
0
= λ/µ.
For improving the accuracy of approximation for
L =
m
k=1
L
k
, the modification (13) for L is used. In
Table 1, the
ˆ
L and the simulation are listed for two
retrial time distributions MER
3
(0.0740741;
4
3m
,
10
3m
)
and CE
3,1
(0.007773;
0.146991
m
,
1.188568
m
), where
MER
k
(p;ν
1
,ν
2
) = pErlang(k, ν
1
) + (1
p)Erlang(k, ν
2
) is the mixture of two Erlang
distributions Erlang(k,ν
i
), i = 1,2 of order k and
CE
k, j
(p;ν
1
,ν
2
) the distribution whose Laplace-
Stieltjes transform is of the form
f
(s) = p
ν
1
ν
1
+ s
k
ν
2
ν
2
+ s
j
+(1 p)
ν
2
ν
2
+ s
j
.
Table 1: L in M/M/5 retrial queue (ρ = 0.8).
Retrial Time m
ˆ
L Sim(c.i.)
MER
3
0.1 2.622 2.523(±0.051)
C
2
V
= 0.5 1.0 4.922 4.961(±0.084)
5.0 16.01 16.06(±0.25)
10.0 30.16 30.24(±0.40)
CE
3,1
0.1 2.785 2.673(±0.054)
C
2
V
= 5.0 1.0 5.439 5.534(±0.090)
5.0 16.79 17.00(±0.26)
10.0 30.97 31.15(±0.46)
4 RETRIAL QUEUE WITH
IMPATIENT CUSTOMERS
Consider an M/M/c/K retrial queue with impatient
customers. The impatience of customers is governed
by the persistence function {b
k
,k = 1,2, ·· ·}, where
b
k
is the probability that after the kth attempt fails, a
customer will make the (k + 1)st one. The retrial time
is exponential whose rate may depend on the number
of failures to enter the service facility. Let γ
k
be the re-
trial rate of the customer that has experienced block-
ing k times. For a technical reason, the number of
retrials of a customer from orbit is limited by m, that
is, b
k
= 0 for k m + 1. Since b
k
= 0 for k m + 1,
it can be easily seen that the system is always stable.
Shin and Moon (2011b) proposed an approximation
for this system as follows.
Let X
0
be the number of customers in service fa-
cility and X
k
the number of customers in orbit who
APPROXIMATE SOLUTIONS FOR SOME ADVANCED MULTISERVER RETRIAL QUEUES
197
have failed k times to enter the service facility. It can
be shown that P
0 j
= P(X
0
= j), 0 j c satisfies the
balance equation of the a birth-and-death process with
birth rates {λ
j
}
K1
j=0
and death rates {µ
j
}
K
j=1
, where
µ
j
= min( j, c)µ and λ
j
can be expressed in terms of
R
k j
that is the proportion of returning customers of
type k who find the service facility of state j as
λ
j
= λ + λ
K
i= j+1
λ
i1
µ
i
!
m
k=1
k1
i=1
b
i
R
iK
!
b
k
R
k j
,
and R
k j
(0 j K,1 k m) is approximated by
R
k j
Z
0
q
K j
(t)γ
k
e
γ
k
t
dt = γ
k
˜q
K j
(γ
k
).
In Table 2, approximation results (App.) are com-
pared with the simulation results (Sim.) with half
length of 95% confidence interval (c.i.) for L
0
, mean
number L
Orbit
of customers in orbit and the block-
ing probability P
B
= P
0c
in the system with γ
k
= γ,
k = 1,2,· ·· , m.
Table 2: M/M/5/7 retrial queue with µ = 1.0, ρ = 1.2.
(b
k
= 0.85
k1
, k = 1,2,3, b
k
= 0, k 4)
γ L
0
L
Orbit
P
B
0.1 App. 6.345 61.42 0.593
Sim. 6.333 60.97 0.589
c.i. ±0.003 ±0.207 ±0.001
1.0 App. 6.252 5.997 0.567
Sim. 6.141 5.835 0.541
c.i. ±0.005 ±0.024 ±0.002
5.0 App. 5.881 1.108 0.481
Sim. 5.744 1.065 0.448
c.i. ±0.005 ±0.003 ±0.001
5 RETRIAL QUEUE WITH
MULTICLASS OF CUSTOMERS
Consider an M/M/c retrial queue with m different
types customers. Let λ
i
, µ
i
and γ
i
denote the arrival
rate, service rate and retrial rate of type i customers
(i-customers), respectively. Assume ρ =
m
i=1
λ
i
i
< 1
for stability of the system. Shin and Moon (2011c)
proposed an approximation for this system as fol-
lows. Let C
i
and N
i
be the number of i-customers
being served and in orbit in stationary state, respec-
tively and set C
C
C = (C
1
,··· ,C
m
), N
N
N = (N
1
,··· ,N
m
).
Let K =
c
i=0
K (i), where K (i) = {(k
1
,··· ,k
m
)
Z
m
+
:
m
j=1
k
j
= i} and Z
+
= {0,1,2,···}. It can be
shown that π(k
k
k) = P(C
C
C = k
k
k), k
k
k K satisfies π
π
πQ = 0,
where π
π
π = (π(k
k
k),k
k
k K ) and Q has same form as the
generator of level dependent quasi-birth-and-death
(LDQBD) process with finite level which describes
Table 3: M/M/3 retrial queue with m = 2(ρ = 0.8).
Γ P
B
L
1
L
2
0.5 App 0.5584 15.301 9.5116
Sim 0.5630 15.361 9.5189
c.i. ±0.0025 ±0.1562 ±0.1027
1.0 App 0.5603 8.3544 5.4528
Sim 0.5673 8.3597 5.4162
c.i. ±0.0025 ±0.1332 ±0.0693
5.0 App 0.5725 2.7856 2.1833
Sim 0.5923 2.7696 2.1402
c.i. ±0.0036 ±0.0673 ±0.0424
10.0 App 0.5832 2.0811 1.7624
Sim 0.6055 2.0321 1.7041
c.i. ±0.0034 ±0.0470 ±0.0324
the ordinary M/M/c/c queue with m classes of cus-
tomers in which the arrival rate a
i
(k
k
k) of i-customer
depends on the system state k
k
k K and service rate is
µ
i
, 1 i m. The arrival rate a
i
(k
k
k) is given by
a
i
(k
k
k) = λ
i
+
γ
i
L
i
R
i
(k
k
k)
π(k
k
k)
,i = 1,2,··· ,m,
where L
i
= E[N
i
], P
B
= P(C
C
C K (c)), R
B
(i) =
k
k
kK (c)
R
i
(k
k
k) and R
i
(k
k
k) is the proportion of return-
ing customers of i-customers from orbit who find the
service facility in state k
k
k. It can be shown that
L
i
=
λ
i
P
B
γ
i
(1 R
B
(i))
, i = 1, 2,· ·· ,m.
Under under the approximation assumption that the
service facility behaves like a LDQBD process with
generator Q and is independent of the retrials, R
i
(k
k
k)
is approximated by
R
i
(k
k
k)
γ
i
P
B
j
j
jK (c)
π(j
j
j)[(γ
i
I Q)
1
]
j
j
jk
k
k
, i = 1, 2,· ·· ,m
and a
i
(k
k
k) is calculated by iteration.
Let Γ =
m
i=1
γ
i
, β
i
=
γ
i
Γ
, 1 i m and β
β
β =
(β
1
,··· ,β
m
). It can be seen that multiclass M/M/c
retrial queue converges to the ordinary M/M/c queue
with discriminatory random order service (DROS)
with the selecting probability β
β
β = (β
1
,··· ,β
m
) de-
noted by M/M/c/DROS(β
β
β) as Γ tends to infinity for
fixed β
β
β. For the queue with DROS discipline, see
(Kim et al., 2011). From this observations, a refine-
ment of L
i,App
(Γ) is proposed by
ˆ
L
i
(Γ) = L
i,App
(Γ) + (L
i,M/M/c/DROS(β
β
β)
L
i,App
(Γ
)),
where L
i,M/M/c/DROS(β
β
β)
is the mean number of i-
customers in queue for M/M/c/DROS(β
β
β) and Γ
is
large enough so that the variation of L
i,App
(Γ) is neg-
ligible for Γ Γ
.
Table 3 lists the approximation results and simu-
lation ones with 95% confidence interval (c.i.) for P
B
ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems
198
and L
i
in M/M/3 retrial queue with m = 2 classes of
customers, µ
µ
µ = (1.0, 2.0), β
β
β = (0.2, 0.8) and the ra-
tio of arrival rates α
α
α = (0.3,0.7), where α
i
=
λ
i
m
i=1
λ
j
,
1 j m.
ACKNOWLEDGEMENTS
The first and second authors were supported by Ba-
sic Research Program through the National Research
Foundation of Korea (NRF) funded by the Ministry
of Education, Science and Technology, Grant Num-
ber 2009-0072282 and 2010-008957, respectively.
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199