APPROXIMATE SOLUTIONS FOR SOME ADVANCED

MULTISERVER RETRIAL QUEUES

Yang Woo Shin

and Dug Hee Moon

Department of Statistics, Changwon National University, 641-773, Changwon, Republic of Korea

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

 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/(sµ) < 1. Then 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 + iµ + 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 + cµ)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(C = i), N

= E[N|C = i], 0 ≤ i ≤ c.

Summing over j in (1) yields

(λ

+ µ

= λ

i−1

+ µ

i+1

, 0 ≤ i < c, (3)

where µ

= iµ, P

−1

= 0 and

= λ + γN

, 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

}

c−1

j=0

and death rates

{µ

}

j=1

and is given by

= P

∏

k=1



k−1



, i = 1, 2,· ·· ,c (5)

with

1 +

∑

i=1

∏

k=1



k−1



−1

Let u

= P(N = j), j ≥ 0. Summing over i in (1) and

(2), we have that

( j + 1)γu

j+1

− jγu

= 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 =

, (7)

where λ

= λ + γN

. Let R

be the proportion of re-

turning customers from orbit who ﬁnd the service fa-

cility in state i, that is,

γN

∑

j=0

γN

, 0 ≤ i ≤ c. (8)

It can be seen from (7) and (8) that

1 − γR

We have from (4) and (8) that

= λ +

·· ·λ

i+1

·· ·µ

, 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

}

c−1

i=0

and

death rates {µ

}

i=1

and is independent of the retrials.

Let Q be the generator of the birth-and-death pro-

cess with birth rates {λ

}

c−1

i=0

and death rates {µ

}

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

, 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

is approximated by the fraction

of returning customers from orbit that ﬁnd the service

facility in state i, that is,

≈

∞

(t)γe

−γt

dt = γ ˜q

(γ), 0 ≤ i ≤ c. (10)

Remark. The approximation of R

in (10) is the

same as that in Fredericks and Reisner (1979).

Once an approximation of 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

as follows

= λ+λ

·· ·λ

c−1

i+1

·· ·µ

γ ˜q

(γ)

1 − γ ˜q

(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

Let λ

λ = (λ

,··· ,λ

s−1

) and denote by g(λ

λ) the right

hand side of the equation (11) and deﬁne a sequence

of λ

(n)

= (λ

(n)

,··· ,λ

(n)

s−1

) by

(n)

= g(λ

(n−1)

), n = 1, 2,· ·· (12)

with λ

(0)

= (λ,··· , λ). Repeating the successive sub-

stitution (12), we choose λ

(n)

as an approximation of

λ if it satisﬁes ||λ

(n)

− λ

(n−1)

∞

< ε for given toler-

ance ε > 0.

Numerical experiments provide that approxima-

tions for L

is the same as the exact result L

For improving the accuracy of approximation for L =

E[N], a modiﬁcation of approximation for L is pro-

vided as follows

L(m) = L

Appr

(m) + (L

M/M/c

− L

Appr

∗

)), (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,Γ

Γ). For

stability of the system ρ =

cµ

< 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,Γ

Γ). Let

be the number of customers at service facility

and X

the number of customers in orbit whose re-

trial phase is of i in stationary state and L

= E[X

0 ≤ i ≤ m. Then, it can be shown that π

= P(X

= j),

0 ≤ j ≤ c satisﬁes the balance equation of the a birth-

and-death process with birth rates {λ

}

c−1

j=0

and death

rates {µ

}

j=1

, where µ

= jµ and λ

can be expressed

in terms of R

that is the proportion of returning cus-

tomers from orbit who ﬁnd the service facility in state

j as

= λ + λ

∏

i= j+1

i−1

1 − R

, 0 ≤ j < c.

Under the approximation assumption that the service

facility behaves like a birth-and-death process with

birth rates {λ

}

c−1

j=0

and death rates {µ

}

j=1

and is

independent of the retrials, R

is approximated as the

probability that a returning customer ﬁnds the service

facility in state j, that is,

≈

∞

c j

(t)α

αexp(Γ

Γt)Γ

dt, 0 ≤ j ≤ c,

where q

i j

(t) is the transition probability of Q.

Then, λ

is calculated by successive substitution.

Numerical experiments provide that approximations

for L

is the same as the exact result L

= λ/µ.

For improving the accuracy of approximation for

L =

∑

k=1

, the modiﬁcation (13) for L is used. In

Table 1, the

L and the simulation are listed for two

retrial time distributions MER

(0.0740741;

)

and CE

3,1

(0.007773;

0.146991

1.188568

), where

MER

(p;ν

,ν

) = pErlang(k, ν

) + (1 −

p)Erlang(k, ν

) is the mixture of two Erlang

distributions Erlang(k,ν

), i = 1,2 of order k and

k, j

(p;ν

,ν

) the distribution whose Laplace-

Stieltjes transform is of the form

∗

(s) = p



+ s





+ s



+(1− p)



+ s



Table 1: L in M/M/5 retrial queue (ρ = 0.8).

Retrial Time m

L Sim(c.i.)

MER

0.1 2.622 2.523(±0.051)

= 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)

3,1

0.1 2.785 2.673(±0.054)

= 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 = 1,2, ·· ·}, where

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

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

= 0 for k ≥ m + 1. Since b

= 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

be the number of customers in service fa-

cility and X

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

= j), 0 ≤ j ≤ c satisﬁes the

balance equation of the a birth-and-death process with

birth rates {λ

}

K−1

j=0

and death rates {µ

}

j=1

, where

= min( j, c)µ and λ

can be expressed in terms of

k j

that is the proportion of returning customers of

type k who ﬁnd the service facility of state j as

= λ + λ

∏

i= j+1

i−1

∑

k=1

k−1

∏

i=1

k j

and R

k j

(0 ≤ j ≤ K,1 ≤ k ≤ m) is approximated by

k j

≈

∞

K j

(t)γ

−γ

dt = γ

˜q

K j

(γ

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

, mean

number L

Orbit

of customers in orbit and the block-

ing probability P

= P

in the system with γ

= γ,

k = 1,2,· ·· , m.

Table 2: M/M/5/7 retrial queue with µ = 1.0, ρ = 1.2.

= 0.85

k−1

, k = 1,2,3, b

= 0, k ≥ 4)

γ L

Orbit

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

, µ

and γ

denote the arrival

rate, service rate and retrial rate of type i customers

(i-customers), respectively. Assume ρ =

∑

i=1

cµ

< 1

for stability of the system. Shin and Moon (2011c)

proposed an approximation for this system as fol-

lows. Let C

and N

be the number of i-customers

being served and in orbit in stationary state, respec-

tively and set C

C = (C

,··· ,C

), N

N = (N

,··· ,N

Let K = ∪

i=0

K (i), where K (i) = {(k

,··· ,k

) ∈

∑

j=1

= i} and Z

= {0,1,2,···}. It can be

shown that π(k

k) = P(C

C = k

k), k

k ∈ K satisﬁes π

πQ = 0,

where π

π = (π(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

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

k) of i-customer

depends on the system state k

k ∈ K and service rate is

, 1 ≤ i ≤ m. The arrival rate a

k) is given by

k) = λ

π(k

,i = 1,2,··· ,m,

where L

= E[N

], P

= P(C

C ∈ K (c)), R

(i) =

∑

k∈K (c)

k) and R

k) is the proportion of return-

ing customers of i-customers from orbit who ﬁnd the

service facility in state k

k. It can be shown that

(1 − R

(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

is approximated by

k) ≈

∑

j∈K (c)

π(j

j)[(γ

I − Q)

−1

]

, i = 1, 2,· ·· ,m

and a

k) is calculated by iteration.

Let Γ =

∑

i=1

, β

, 1 ≤ i ≤ m and β

β =

(β

,··· ,β

). 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

) de-

noted by M/M/c/DROS(β

β) as Γ tends to inﬁnity for

ﬁxed 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

(Γ) = L

i,App

(Γ) + (L

i,M/M/c/DROS(β

β)

− L

i,App

(Γ

∗

)),

where L

i,M/M/c/DROS(β

β)

is the mean number of i-

customers in queue for M/M/c/DROS(β

β) and Γ

∗

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

ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems

198

and L

in M/M/3 retrial queue with m = 2 classes of

customers, µ

µ = (1.0, 2.0), β

β = (0.2, 0.8) and the ra-

tio of arrival rates α

α = (0.3,0.7), where α

∑

i=1

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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APPROXIMATE SOLUTIONS FOR SOME ADVANCED MULTISERVER RETRIAL QUEUES

199