1 queue

Microelectron. Relish., Vol. 35. No. 6, pp. 915-921. 1995 Copyright fc 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0026-27 14195 $9.50 + .OO

Pergamon 0026-2714(94)00127-8

PROBLEM OF STATISTICAL

INFERENCE QUEUE

FOR HEAVY TRAFFIC IN M/M/l

SUDHA JAIN Department of Statistics, University of Toronto, Toronto, Ontario, Canada MSS 1Al (Received for publication 19 August 1994)

ABSTRACT

In this paper, the upper bounds for waiting time distribution in M/M/l queue under heavy traffic assumption are compared with those based on statistical estimation procedures. Simulation studies indicate that the results based on statistical estimation procedures perform better.

1. Introduction

Kingman (1962) derived the upper bounds for the waiting time distribution for all G/GI/I queues. Kingman’s (1962, 1964) interest was to investigate what sort of results can be expected to hold for heavy traffic system. Later, Marshall (1968) discussed the same problem and obtained bounds for various measures of performance in certain classes of the GI/G/l queue. Marchal (1978) obtained the lower bound for the mean weir in queue dependent upon interarrival and service times moments only, which is similar in characteristics to that of the upper bound discussed by [Kingman (1962) and Marshall (1968)]. Under heavy traffic, the waiting time distribution is approximately negatively exponentially distributed with mean h(oi + 0:)/2(1

- p), where of is the

1 variance of interarrival time, and 0: is the variance of the service time, - IS the mean k 1 of interarrival time, - IS the mean of the service time and p = t cc

is the traffic intensity.

The above result is due to Kingman (1962, 1964). In this paper, statistical estimation procedures for the waiting time in queue under heavy traffic approximations are investigated based on Kingman’s result. In section 2, the preliminary results and notations are given. Section 3, deals the related statistical estimation procedures for developing approximate confidence intervals of the waiting times in M/M/l queue and then the comparisons are made with the corresponding bounds. The simulation studies for the M/M/l queue indicate that results based on statistical estimation procedures pertkm better. 91

916

S. JAIN

2. Preliminary results and notations Let a(t) and g(t) be the density functions for the interarrival and service times respectively. Denote 1 --~ = M e a n o f the interarrival time

1 - - = M e a n o f the service time ~t p = - - = Traffic Intensity ~t 02 = Variance o f the interarrival time

o2s = Variance o f the service time

W n = Waiting time in queue o f the nth customer

The random variable W will represent the steady-state waiting time in queue (p < 1). In a GI/G/I queue with traffic intensity p < 1, the upper bounds of the wait in queue [Marshall (1968) and Kingman (1962)] is given by ~.(o~2 + 0 2)

E(W)<

2(1 - p)

(2.1)

It is interesting to note that the upper bounds of wait in queue depends on only the mean and variance of the interamval and service times and further knowledge of the distribution is not required. The lower bounds for wait in queue based on Marchal's (1978) work is given by Gross and Harris (1985) as follows: ~,2a2 + p(p - 2) E(W) >

2X(I - p)

(2.2)

Remark, The above lower bound in (2.2) is dependent only upon interarrival and service time moments. Further, it is positive if and only if og2 > (2 - p)/~t. The derivation procedure of the lower and upper bound suggests that the upper bound will be sharper for waiting time in queue when p is close to I. M/M/I queue. Since oa2 =

and ag2 = -~-2" Theref°re' the upper b°und °f E(W)

for M/M/I queue from (2.1) is given by E(W)_<

1 + p2 2~.(1 - p) "

(2.3)

Another, lower bound for M/M/I queue from Gross and Hams (1985, page 412) is given by

Queueing

917

E ( W ) > --~ln(l - p2).

(2.4)

It is interesting to note that both upper and lower bounds increase sharply as p goes to 1. The Table 2.1 presents the upper and lower bounds for E ( W ) in M/M/I queue for various value of p. Table 2.1. Upper and lower bounds for E ( W ) in M/M/1 queue

p

Upper bound

Lower bound

0.98

50.0103

3.2948

0.95

20.0263

2.4504

0.90

10.0556

1.8453

0.85

06.7549

1.5082

0.80

05.1250

1.2771

0.75

04.1667

1.1022

0.70

03.5476

0.9619

0.65

03.1263

0.8447

0.60

02.8333

0.7438

3. Statistical Estimation

The waiting time distribution under heavy traffic assumption is approximately negative exponential with mean k(o 2 +~2)/2(1-p) [Kingman (1962, 1964)]. Kingman (1964) remarked that the preceding result is one of the fundamental basis in the development of heavy traffic theory. However, the precise form is quite difficult to obtain and further it is irrelevant to the practical applications. It can be shown that under the above assumption, the sampling distribution of E ( W ) = 0 is given by = ~-n .X2(2n).

(3.1)

Confidence limits for 0, at significance level ct are given by 2T,, Lower Limit = Z2r2(2n )

(3.2)

2r. Upper Limit - X2l _ t~'2(2n)

(3.3)

where T,, = w! + w2 + ...+w,, and Wi is the waiting time of the ith customer. The Tables 3.1 to 3.6 present the 95% confidence interval of the E ( W ) in M/M/I queue based on the simulation studies of the first 35 customers repeated 20, 50, 100, 250, 500, and lO00 times for various values ofp.

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S. JAIN

Table 3.1. 95% Confidence Interval for E(W) in M/M/I queue for the first 35 customers repeated 20 times.

p

W

Lower Limit

Upper Limit

Width of CI

0.98

4.7506

3.2022

7.7773

4.5751

0.95

4.2552

3.0843

7.4911

4.4068

0.90

4.1260

2.7811

6.7547

3.9736

0.85

3.3367

2.2491

5.4626

3.2135

0.80

3.2558

2.1946

5.3301

3.1355

0.75

3.0339

2.0450

4.9668

2.9218

0.70

2.7446

1.8500

4.4932

2.6432

0.65

2.6154

1.7629

4.2817

2.5188

0.60

2.1607

1.4564

3.5373

2.0809

Table 3.2. 95% Confidence intervals for E(W) in M/M/1 queue for the first 35 customers repeated 50 times.

P

W

Lower Limit

Upper Limit

Width of CI

0.98

5.1171

3.9495

6.8943

2.9448

0.95

4.5971

3.5482

6.1937

2.6455

0.90

3.6447

2.8331

4.9105

2.0774

0.85

3.6368

2.8074

4.8999

2.0925

0.80

3.3551

2.5895

4.5203

1.9308

0.75

3.1696

2.4464

4.2704

1.8240

0.70

2.5302

1.9528

3.4089

1.4561

0.65

2.3907

1.8452

3.2210

1.3758

0.60

2.2784

1.7585

3.0697

1.3112

Queueing

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Table 3.3. 95% Confidence Intervals for E(W) in M/M/I queue for the first 35 customers repeated 100 times.

p

W

Lower Limit

Upper Limit

Width of CI

0.98

4.4619

3.7019

5.4838

1.7819

0.95

4.3745

3.6294

5.3764

1.7470

0.90

3.9567

3.2827

4.8629

1.5802

0.85

3.6551

3.0325

4.4922

1.4597

0.80

3.4996

2.8595

4.2306

1.3765

0.75

3.4152

2.8335

4.1974

1.3639

0.70

2.8383

2.3548

3.4883

1.1335

0.65

2.3974

1.9890

2.9465

0.9575

0.60

2.2226

1.8440

2.7316

0.8876

Table 3.4. 95% Confidence intervals for E(W) in M/M/1 queue for the first 35 customers repeated 250 times.

p

W

Lower Limit

Upper Limit

Width of CI

0.98

4.5126

4.0006

5.1285

1.1279

0.95

4.1541

3.6827

4.7209

1.0382

0.90

3.9509

3.5026

4.4901

0.9875

0.85

3.8517

3.4147

4.3774

0.9627

0.80

3.0836

2.7337

3.5044

0.7707

0.75

3.0094

2.6678

3.4200

0.7522

0.70

2.6898

2.3842

3.0564

0.6722

0.65

2.4518

2.1736

2.7864

0.6128

0.60

2.2578

2.0016

2.5659

0.5643

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S. JAIN

Table 3.5. 95% Confidence intervals for E(W) in M/M/1 queue for the first 35 customers repeated 500 times.

p

W

Lower Limit

Upper Limit

Width of CI

0.98

4.2515

3.9072

4.6561

0.7489

0.95

3.9297

3.6065

4.2977

0.6902

0.90

3.7345

3.4273

4.0842

0.6569

0.85

3.3765

3.0987

3.6926

0.5939

0.80

3.1289

2.8715

3.4219

0.5504

0.75

2.8993

2.6562

3.1653

0.5091

0.70

2.6841

2.4358

2.9026

0.4668

0.65

2.4894

2.2846

2.7225

0.4379

0.60

2.4095

2.0566

2.4507

0.3941

Table 3.6. 95% Confidence intervals for E(W) in M/M/I queue for the first 35 customers repeated 1000 times.

p

W

Lower Limit

Upper Limit

Width of CI

0.98

4.3295

4.0729

4.5609

0.4880

0.95

4.0954

3.8528

4.3144

0.4616

0.90

3.8044

3.5789

4.0077

0.4288

0.85

3.4749

3.2690

3.6607

0.3916

0.80

3.3005

3.1049

3.4769

0.3720

0.75

2.9352

2.7613

3.0921

0.3308

0.70

2.7214

2.5601

2.8668

0.3067

0.65

2.4588

2.3129

2.5900

0.2771

0.60

2.2209

2.0892

2.3395

0.2503

Queueing

Concluding Remarks. Kingman (1964) stated that the precision of the approximation may be improved by setting the theory on a more elegant and rigorous mathematical basis. In this paper, we use the statistical estimation procedures to improve the approximation for E(W) in M/M/I queue. Under heavy traffic the results of simulations in Tables 3.1 to 3.6 clearly indicate that the average wait of the customer increases when the P approaches 1. However, there is a significant difference with the bounds given in Table 2.1. Furthermore, it may also be noted that the average wait also increase with the size of queue when the traffic intensity p is nearly one.

Acknowledgement. The author is grateful to NSERC (Canada) for supporting this research.

References. Gross, D. and Harris, C. M. Fundamentals of Queueing Theory. Second ed., John

Wile)', (1985). Heyman, D. P. and Sobel, M. J. Stochastic Models in Operations Research. Vol. 1,

McGraw-Hill, (1982). Kingman, J. F. C. On queues in heavy traffic. J. Roy. Stat. Soc., B 24, 383-392.

(1962). Kingman. J. F. C. The heavy traffic approximation in the theory of queues.

Proceedings of Symposium on Congestion Theory. (ed by W. L. Smith and W. E. Wilkinson, North Carolina Press, Chapl Hill, (1964). Marchal, W.G. Some simpler bounds on the mean queueing time. Oper. Res. 26,

1083-1088. (1978). Marshall, K. T. Some inequalities in queueing. Oper. Res., Vol. 16, 651-665.

(1968).

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