THE ROBUSTNESS OF BLOCKING PROBABILITY IN A LOSS

SYSTEM WITH REPEATED CUSTOMERS

Akira Takahashi, Yoshitaka Takahashi

Graduate School of Commerce , Waseda University

Shinjuku, Tokyo 169-8050, Japan

Shigeru Kaneda, Yoshikazu Akinaga and Noriteru Shinagawa

Network Laboratories , NTT DoCoMo,Inc.

3-5, Hikarinooka, Yokosuka, Kanagawa, 239-8536, Japan

Keywords:

Teletrafﬁc analysis, loss system, repeated customers, Little’s formula.

Abstract:

In this paper, we analyze and synthesize a multi-server loss system with repeated customers, arising out of NTT

DoCoMo-developed telecommunication networks. We ﬁrst provide the numerical solution for a Markovian

model with exponential retrial intervals. Applying Little’s formula, we derive the main system performance

measures (blocking probability and mean waiting time) for general non-Markovian models. We compare the

numerical and simulated results for the Markovian model, in order to check the accuracy of the simulations.

Via performing extensive simulations for non-Markovian (non-exponential retrial intervals) models, we ﬁnd

robustness in the blocking probability and the mean waiting time, that is, the performance measures are

shown to be insensitive to the retrial intervals distribution except for the mean.

1 INTRODUCTION

When the service system becomes extremely con-

gested, a lot of customers cannot receive immediate

service. Some of them may give up the service to

leave the system, while others may stay in the sys-

tem and retry their requests. This behavior of re-

peated customers leads to an additional load on the

system and worsens its congestion. The importance

of repeated requests on the performance of the ser-

vice system was pointed out in the late 1940, and

many researches have been performed since then. Pi-

oneering studies on the multi-server loss system with

repeated calls brought some kind of positive expres-

sions of performance measures (See (Falin and Tem-

pleton, 1997), (Artalejo and Pozo, 2002), and (Uda-

gawa and Miwa, 1965)). However, they are not neces-

sarily convenient to calculate performance measures.

Retrial queuing models including one discussed here

are usually very complicated for queuing analysis and

its results are not always suitable to numerical calcu-

lation. Many authors reported numerical approaches

of approximation and truncation methods. For details

on the numerical approaches, readers are referred to

(Artalejo and Pozo, 2002) and (Stepanov, 1999).

Most of them assume that the time intervals be-

tween repeated attempts are mutually independent

and exponentially distributed. However, affected by

many factors and circumstances, customers’ behav-

ior in repeating is so complex that these assumptions

may lead to a risky assessment. There is necessity for

generalization of the retrial interval distribution.

This assumption of exponential retrial intervals is

a kind of simpliﬁcation for queuing analysis. There

is no guarantee that repeating customers behave in

such a manner. Under this assumption, the number

of repeated requests emerging in a unit time changes

by the state of the retrial queue (See (Artalejo et al.,

2001)). There is another type of retrial queuing model

in which retrial rate is constant. In this type of model,

blocked customers who want to repeat must wait in

line and only the customer at the head of the line can

retry to hunt, if any, an idle server. It has a wide

range of applications like communication protocols.

However, still there are systems more appropriately

modeled by the classical type. On the constant re-

trial policy, there are fruitful investigations of mod-

els with single-server non-exponential retrial intervals

like (G

´

omez-Correl and Ramalhoto, 1999). However,

it remains open problem to investigate the effect of

the retrial times distribution on the performance of the

system. Customers’ behavior in repeating is expected

to be highly complex and it may be risky or inefﬁ-

cient to build and operate the system upon the results

of exponential assumption. Thus, one ﬁnds it neces-

sary to study sensitivity and robustness of the retrial

60

Takahashi A., Takahashi Y., Keneda S. and Akinaga Y. (2005).

THE ROBUSTNESS OF BLOCKING PROBABILITY IN A LOSS SYSTEM WITH REPEATED CUSTOMERS.

In Proceedings of the Second International Conference on e-Business and Telecommunication Networks, pages 61-66

DOI: 10.5220/0001416300610066

Copyright

c

SciTePress

time distribution

The main goal of this paper is to investigate the ro-

bustness (insensitivity) property between the perfor-

mance measure and the retrial time distribution in a

loss system with repeated customers seen in an NTT

DoCoMo developed telecommunication network.

The rest of the paper is organized as follows. Sec-

tion 2 gives teletrafﬁc analysis of the retrial model.

The main performance measures of practical inter-

est are then derived. In Section 3, simulation results

of non-exponential (deterministic/ two-stage Erlang/

two-stage hyper-exponential) retrial models are com-

pared to ﬁnd robustness of the system.

2 NUMERICAL RESULTS

2.1 Model Description

Consider the following loss system with repeated

calls : (1) There are c servers in parallel; (2) Cus-

tomers’ service times, which are identical and inde-

pendent from one another, are exponentially distrib-

uted with rate µ; (3) Customer arrivals follow a Pois-

son process of rate λ; (4) Customers who ﬁnd all sev-

ers busy at their arrival epoch choose either to repeat

their requests with probability p or to give up the ser-

vice with probability (1−p); (5) When they decide to

repeat, customers wait in the retrial queue for a ran-

dom time called ”retrial interval”, which is exponen-

tially (generally) distributed with parameter γ in Sec-

tion 2.2( Section 3) before making repeated requests;

(6) Retrial customers who ﬁnd again all servers busy

choose either to waint in the retrial queue and repeat

their requests with probability p or to stop repeating

and leave the system with probability (1 − p); (7)

Give-up customers leave the system immediately.

We introduce following notations. Suppose the ex-

istence of the stationary state, the state of the sys-

tem is characterized by (1) the number of the busy

servers and (2) the number of the customers waiting

to make a repeated attempt. The system will be said

to be in state (i, j) , if i servers busy and j customers

waiting to repeat. If there are c servers in the sys-

tem then the system is somewhere in the state space

{0, 1, ··· , c} ×{0, 1, ···} . Let π

i,j

denote the prob-

ability that the system is in state (i, j) from now on.

2.2 Calculation of the Stationary

Distribution

By focusing on the possible state transition in a

minute time ∆t, we get the state-transition probabili-

ties as shown in Table 1. Figure 1 illustrates the state-

transition diagram of this model.

Table 1: State-transition probabilities

state-transition probability

(i + 1, j)

↑ (0 ≦ i ≦ c − 1, 0 ≦ j) λ∆t + o(∆t)

(i, j)

(i + 1, j − 1)

տ (0 ≦ i ≦ c − 1, 1 ≦ j) jγ∆t + o(∆t)

(i, j)

(i, j)

↓ (1 ≦ i ≦ c, 0 ≦ j) iµ∆t + o(∆t)

(i − 1, j)

(c, j) → (c, j + 1)

λα∆t + o(∆t)

(0 ≦ j)

(c, j − 1) ← (c, j)

jγ(1 − α)∆t

(1 ≦ j) +o(∆t)

(c, j − 1) ← (c, j)

jγ(1 − α)∆t

(1 ≦ j) +o(∆t)

By Table 1, the state-equilibrium equations are ex-

pressed as below.

(λ + jγ)π

0,j

= µπ

1,j

.

(λ + iµ + jγ)π

i,j

= λπ

i−1,j

+ (j + 1)γπ

i−1,j

+ (i + 1)µπ

i+1,j

(1 ≦ i ≦ c − 1).

(λα + cµ + jγ(1 −α))π

c,j

.

= λπ

c−1,j

+ γπ

c−1,j

+ λαπ

c,j−1

+ (j + 1)γ(1 − α)π

c,j+1

.

Our research purpose here is to study the effect

of the retrial interval distribution and the existence

THE ROBUSTNESS OF BLOCKING PROBABILITY IN A LOSS SYSTEM WITH REPEATED CUSTOMERS

61

Figure 1: State-transition diagram.

of robustness. To this end, we adopt an simple ap-

proximation method of replacing the inﬁnite space for

customers waiting to repeat requests by a ﬁnite num-

ber k. It is an extension of the way to calculate the

steady-state probabilities introduced in (Hashida and

Kawashima, 1979) and closely explained in (Falin

and Templeton, 1997), so we only show the outline of

the algorithm. Here, k is assumed to be a sufﬁciently

large positive integer so that the overﬂow probability

is small enough to be ignored. From a ﬁnite capacity

argument,

for 1 ≦ j ≦ k − 1,

(λ + jγ)π

0,j

= µπ

1,j

, (1)

(λ + iµ + jγ)π

i,j

= λπ

i−1,j

+(j + 1)γπ

i−1,j

+ (i + 1)µπ

i+1,j

(1 ≦ i ≦ c − 1), (2)

(λα + cµ + jγ(1 −α))π

c,j

= λπ

c−1,j

+ γπ

c−1,j

+ λαπ

c,j−1

+(j + 1)γ(1 − α)π

c,j+1

. (3)

For j = k,

(λ + kγ)π

0,k

= µπ

1,k

, (4)

(λ + iµ + kγ)π

i,k

= λπ

i−1,k

+ (i + 1)µπ

i+1,k

(1 ≦ i ≦ c − 1), (5)

(cµ + kγ(1 − α))π

c,k

= λπ

c−1,k

+ λαπ

c,k−1

. (6)

These recurrence equations enable us to compute

the stationary distribution via the following steps.

(I) Take the appropriate k and introduce auxiliary

variables φ

i,j

, pi

i,j

/π

0,k

.

(II) By deﬁnition , φ

0,k

= 1.

From (4), φ

1,k

can be determined.

From (5), one can get φ

i,k

(i = 2, 3, ··· , c)

sequentially.

(III) Equations (1) and (2) for i = 1, ··· , c − 1

constitute a set of c equations with c + 1

unknown variables φ

0,k

, φ

1,k

, ··· , φ

c−1,k

, φ

c,k

. Thus, with φ

c,k

obtained by (6), one

ﬁnds the set of equations become solvable .

Hence, (3) gives φ

c,k−2

D

(IV ) Operating steps (I), (II), and (III)

repeatedlyC we get all of the φ

i,j

.

The normalization condition;

P

c

i=0

P

k

j=0

φ

i,j

= 1/φ

0,k

settles π

0,k

and φ

i,j

× π

0,k

gives π

i,j

.

(V ) By repeating from (I) to (IV ) with k plus 1

until the value of π

c,k

becomes less than 10

−10

,

π

i,j

can be calculated with an accuracy enough

for our purpose.

2.3 Performance Measures

We are now in a position to derive the performance

measures of the system.

Time congestion (B

T

)

Letting B

T

be the time congestion, so called, the

probabilities that all the servers are busy, we have

B

T

=

k

X

j=0

π

c,j

.

Blocking probability (B)

When they blocked due to all servers busy, customers

can wait for some random time and retry. After sev-

eral retrials, some of them may give up the service

demand, and leave the system. Here, we deﬁne the

ICETE 2005 - WIRELESS COMMUNICATION SYSTEMS AND NETWORKS

62

blocking probability B as the probability that cus-

tomers ﬁnally leave without getting served due to suc-

cessive blockings.

To the best of authors’ knowledge, there are few in-

vestigations for a general retrial model. Here, apply-

ing Little’s formula (Little, 1961) enables us to prove

the following proposition.

Proposition 1 Consider a general retrial queuing

system which has input with rate λ, service with rate

µ and retrial with rate γ. Customers who try to re-

ceive a service and get blocked due to all servers busy,

choose either to repeat their requests with probability

p or to stop repeating and leave the system with prob-

ability (1 − p). Blocking probability B, that is, the

probability that arriving customers ﬁnally leave the

system with not receiving the service, is expressed by

B = 1 −

1

ρ

C.

Here, ρ denotes trafﬁc intensity deﬁned by λ/µ and

C stands for the mean number of busy servers on the

stationary condition. See Appendix 1 for the proof.

It should be noted that Proposition 1 has a different

expression on the blocking probability from that in

(Hashida and Kawashima, 1979), where the P AST A

(P oisson Arrivals See T ime Averages) property

is heuristically used to provide an approximation. Our

expression on the loss probability shown in Proposi-

tion 1 is exact (not approximate).

Mean waiting time (W q)

Denote by Wq the mean waiting time, namely, the

mean elapsed time from a customer’s arrival epoch

until the epoch where the customer gets served or

stops repeating without receiving its service to leave

the system.

Like B above, W q is also derived from Little’s for-

mula and its relation to other parameters is preserved

under more general situation. So we ﬁnd the follow-

ing proposition.

Proposition 2 Consider a general retrial queuing

system which has input with rate λ, service with rate

µ and retrial with rate γ. Customers who try to re-

ceive a service and get blocked due to all servers busy,

choose either to repeat their requests with probability

p or to stop repeating and leave the system with prob-

ability (1 − p). The mean waiting time W q, that is,

the time that customers have to spend on average until

they ﬁnally get served or decide to stop repeating and

leave, is expressed by

W q =

K

λ

.

K is the mean number of customers in the retrial area

in the steady state. See Appendix 2 for the proof.

3 SIMULATION RESULTS

In the previous section, we get the numerical solution

of the loss system with exponential retrial intervals.

Next, we change the assumption about retrial. In this

section we compare performance measures between

the exponential retrial interval model and the mod-

els with non-exponential retrial intervals. Even un-

der the exponential retrial interval assumption, multi-

server property involves great complicity and analyt-

ical solutions are obtained only a few special cases

like (Falin and Templeton, 1997) and (Choi and Kim,

1998). So we employ computer simulation to esti-

mate the performance measure of non-exponential re-

trial interval models. The assumptions for simulation

are all the same with those for numerical calculation

introduced in Section 2 except for the distribution of

retrial intervals. It assumes a Poisson arrival of cus-

tomers with rate λ and an identically independently

distributed exponential service time with rate µ.

On the distribution of retrial intervals, in this pa-

per we take four different models; the exponential re-

trial interval model (Exp model) the constant retrial

interval model (D model), the 2-stage Erlang distrib-

ution model (E2 model), and the 2-stage hyper expo-

nential distribution model (H2 model). Among H2

models, we also have three different types whose co-

efﬁcient of variation (C

X

) of the retrial interval dis-

tribution is lager than 1, equal to 1, or smaller than

1. In other words, the variance of the retrial in-

terval distribution is large, equal or small in com-

parison to its mean. H2(C

X

=

√

2), H2(C

X

=

√

20) and H2(C

X

=

√

200) denote the model with

hyper-exponential retrial intervals whose C

X

equals

to

√

2,

√

20 and

√

200, respectively .

Through this section, τ , µ/γ is used for the indi-

cator of the mean retrial interval and ρ , λ/µ for the

trafﬁc offered to the whole system.

In simulation, c(= the number of servers) is set to

10, µ 0.01and p 5/6, which means the service time

average is 100 and under the condition of successive

blocking customers continue to repeat 5 times on av-

erage. An individual simulation results (expressed as

points in each ﬁgure) is based on 50 runs(approx. 5

hours on IBM Thnkpad PC).

First, the accuracy of the simulation should be in-

vestigated. Figure 2 shows the blocking probability

B by numerical calculation and simulation with the

mean retrial time τ = 1. As seen in Figure 2, we can-

not see signiﬁcant difference between our numerical

and simulation results. Therefore, our simulation re-

sults are very accurate. The accuracy of simulation is

conﬁrmed on other performance measures.

Now that we see the accuracy of the simulation,

comparisons are performed when the mean retrial in-

terval τ is 0.01, 1, and 100.0, which corresponds to

THE ROBUSTNESS OF BLOCKING PROBABILITY IN A LOSS SYSTEM WITH REPEATED CUSTOMERS

63

Figure 2: The blocking probability B by numeric al

calculation and simulation.

the situations that repeated requests arises 100 times

sooner than the service time on average, that they

arise with the interval as long as the service time on

average, and that they arise after a time 100 times

longer than the service time on average.

Figures 3 and 4 show the relationship of the block-

ing probability B and the mean waiting time W q to

the trafﬁc intensity ρ. One ﬁnds the retrial interval

distribution makes little difference.

4 CONCLUSION

We have analyzed and synthesized a multi-server

loss system with repeated customers, arising out

of NTT DoCoMo-developed telecommunication net-

works. We have ﬁrst provided the numerical solution

for a Markovian model with exponential retrial in-

tervals. Applying Little’s formula, we have derived

the main system performance measures (blocking

probability and mean waiting time) for general non-

Markovian models. We have compared the numeri-

cal and simulated results for the Markovian model, in

order to check the accuracy of the simulations. Via

performing extensive simulations for non-Markovian

(non-exponential retrial intervals) models, we have

found robustness in the blocking probability and the

mean waiting time, that is, the performance measures

have been shown to be insensitive to the retrial inter-

vals distribution except for the mean.

It is left for future work to investigate the robust-

ness for a more general (e.g., a general service time

distribution) model.

Figure 3: The blocking probability B versus the trafﬁc

intensity ρ.

ACKNOWLEDGEMENT

The present research was partially supported by a

Grant-in-Aid for Scientiﬁc Research (C) from Japan

Society for the Promotion of Science under Grant No.

1458049.

REFERENCES

Artalejo, J., G

´

omez-Correl, A., and Neuts, M. (2001).

Analysis of multiserver queues with constant retrial

rate. European Journal of Operational Research,

135:569–581.

Artalejo, J. and Pozo, M. (2002). Numerical calculation

of the stationary distribution of the main multiserver

retrial queue. Annals of Operations Research, 116:41–

56.

Choi, B. and Kim, Y. (1998). The M/M/c retrial queue with

geometric loss and feedback. Computers and Mathe-

matics with Applications, 36:41–52.

Falin, G. and Templeton, J. (1997). Retrial Queues. Chap-

man and Hall, London, 1st edition.

G

´

omez-Correl, A. and Ramalhoto, M. (1999). The station-

ary distribution of a markovian process arising in the

theory of multiserver retrial queueing systems. Math-

ematical and Computer Modelling, 30:141–158.

Hashida, O. and Kawashima, K. (1979). Buffer behav-

ior with repeated calls. The IECE Transactions, J62-

B:222–228.

Little, J. D. C. (1961). A proof for the queuing formula: L =

λ W. The Journal of the Operations Research Society

of America, 9:383–387.

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Stepanov, S. (1999). Markov model with retrials:the cal-

culation of stationary performance measures based on

the concept of truncation. Mathematical and Com-

puter Modelling, 30:207–228.

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Figure 4: The mean waiting time W q versus the trafﬁc

intensity ρ.

APPENDICES

Appendix 1

Proof of Proposition 1

We restrict ourselves to the sub-system only composed

of c servers. We apply Little’s formula [ L = λW ] (Little,

1961) to this sub-system.

Let λ

′

and

C respectively denote the sub-system

throughput and the mean number of busy servers. By Lit-

tle’s formula, we have

C = λ

′

1

µ

, (A.1)

now that

C [the mean number of customers in the sub-

sytem] corresponds to L, λ

′

[the effective arrival rate of the

sub-system] corresponds to λ, and 1/µ [the mean sojourn

time in the sub-system] corresponds to W .

The blocking probability B is deﬁned as the probabil-

ity that an arriving customer cannot ﬁnally receive its ser-

vice, however often it may repeat the retrial process [being

blocked, waiting, and retrying].

Since on average λ customers arrive at the system in unit

time, the mean number of customers who leave the system

without being served is given by λB. The mean number of

customers who receive their services is obtained as

λ(1 − B) = λ

′

. (A.2)

Substituting (A.2) into (A.1), and solving for B we ﬁ-

nally get

B = 1 −

1

ρ

C.

Appendix 2

Proof of Proposition 2

We restrict ourselves to the sub-system only composed

of the retrail queue (with a ﬁnite capacity of k customers).

We apply Little’s formula [ L = λW ] to this sub-system.

Let S and

K denote the mean number of retrials and the

mean number of customers in the retrial queue, respectively.

Since on average λ customers arrive at the system in time

unit, then each one of them go through the retrial queue S

times on average. That is, the mean number of customers

who go to the retrial queue in time unit equals to Sλ. By

Little’s formula, we have

K =Sλ

1

µ

(A.3)

now that

K [the mean number of customers in the sub-

sytem] corresponds to L, Sλ [the effective arrival rate of the

sub-system] corresponds to λ, and 1/µ [the mean sojourn

time in the sub-system] corresponds to W . From (A.3), we

have

S =

Kγ

λ

(A.4)

Since the mean retrial interval is 1/µ and customers re-

peat their requests S times on average, then the mean waint-

ing time W q is

W q = S

1

γ

(A.5)

Substituting (A.4) to (A.5), we get

W q =

K

λ

.

THE ROBUSTNESS OF BLOCKING PROBABILITY IN A LOSS SYSTEM WITH REPEATED CUSTOMERS

65