2 queue with heterogeneous servers

Operations Research Letters 10 (1991) 99-101 North-Holland

March 1991

Confidence interval for M / M / 2 queue with heterogeneous servers Sudha Jain * and J.G.C. Templeton Department of Industrial Engineerin& Unioersity of Toronto, Toronto, Canada M5S 1A4 Received May 1989 Revised March 1990

This paper derives the confidence interval formula for the parameters of M / M / 2 m a x i m u m likefihood estimate; M / M / 2

queue with heterogeneous servers.

queue; ordered heterogeneous servers

1. Introduction The problem of estimating traffic intensity for M / M / 1 queue using maximum likelihood principles was discussed by Clarke [1]. Lilliefors [4] examined the problem of finding the confidence interval for the actual traffic intensity. He used the estimates of traffic intensity to obtain the confidence interval for the expected number of units in the system. The precision of the estimates as measured by their confidence limits were obtained as a function of the number of arrivals. Recently, Dave and Shah [2] obtained the estimates of the parameters for M / M / 2 queue with ordered heterogeneous servers. They estimated the arrivals and service rates based on maximum likelihood principles assuming the queue in equilibrium. The derivation of the confidence intervals requires the familiarity of the distribution properties of the estimators. In this paper, Lilliefors' [4] technique is employed to derive the confidence intervals for the parameters of M / M / 2 queue with heterogeneous servers.

2. M / M / 2

queue with heterogeneous servers

The assumptions and notations in this paper are those as given in Dave and Shah [2]. In this queueing system, customers wait in line according to their arrival. When both servers are idle, the faster server is scheduled for service before the slower one. Notations. Observations begin with m 0 customers and continue for time T. During the period T denote m u - number of arrivals, number of departures, T~ -= amount of time during which both servers are idle,

m d

~

Trb ----amount of time during which only the faster server is busy, Tsb ~ amount of time during which both servers are busy. * Presently at Department of Mathematics and Statistics, Queen's University, Kingston, Canada K 7 L 3N6. 0167-6377/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

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When only the faster server is busy, let m fu = number of arrivals to the partially busy queue, m fd = number of departures. When both servers are busy, let msu = number of arrivals to the queue, m~d - number of departures. Dave and Shah [2] assume T = T~ + Tfb + T~b, ignoring the possibility that only the slow server m a y be busy. This assumption is valid if, when both servers have been busy and no customers are waiting, the customer with the slow server immediately transfers to the fast server if the fast server completes service and becomes idle. Assuming that the queue is in equilibrium, the m a x i m u m likelihood estimates for the mean arrival rate h and two unequal mean service rates gl and g2 (gl > g2) are given by (Dave and Shah [2])

(1) (2) (3)

X = mu/T, ~1 = mfd/Tf,o, ~2 = m s J T s , o - ~ .

3. Confidence intervals The estimate of traffic intensity p is given by

(4)

# = ##,- -+- - ~ = ~ , where

ti=~1+ti2. F r o m equation (3), we get

(5) Thus, t3 is given by m u/T

(6)

where T = m u/~. Consider the ratio P =

2 g T~b/ 2 m ~d 2XT/2m u

(7)

Since arrival and service times are independent and 2gT~b and 2XT have chi-square distributions with degrees of freedom 2m~d and 2 m . respectively; then the ratio ~ / p has an F-distribution with 2msd and 2 m , degrees of freedom. Thus, an appropriate probability statement can be written as follows: Pr Fl_~,/2(2msa, 2 m u ) ~ ~ ~ Fa/2(2msa , 2 m . ) 100

= 1 - et.

(8)

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March 1991

Hence, a 100(1 - a)% confidence interval for p = ~//(]~1 --I-/£2) is given by F~/2(msd ' 2rnu ) ~< p ~< F l _ ~ / 2 ( 2 m s d , 2 m u ) .

(9)

The average n u m b e r of customers N in the system is given by Trivedi [5] as follows:

x--2[/A1/Z2(1 + 2 0 )

[

1

]-1

+-f=Tj

(lO)

Let

/'1=a/~2

ifa>l;

then the average n u m b e r of customers is given by E(N)

=

p(1 + a ) ( 1 + ap + p) (1 - p ) ( a + p + p2 + 2 a p + a2#:) "

(11)

It can be shown that E ( N ) is a m o n o t o n i c increasing function of p (0 < p < 1) if a is greater than one since server 1 is faster than server 2. Hence confidence limits for E ( N ) can be written b y substituting lower and u p p e r confidence limits of p using formula (9) in E ( N ) .

Acknowledgement The authors wish to thank a referee for comments.

References [1] A.B. Clarke, "Maximum likelihood estimates in a simple queue", Ann. Math. Statist. 28, 1036-1040 (1957). [2] U. Dave and Y.K. Shah, "Maximum likelihood estimates in M/M/2 queue with heterogeneous servers", J. Oper. Res. Soc. 31, 423-426 (1980). [3] D. Gross and C.M. Harris, Fundamentals of Queueing Theory (second edition), Wiley, New York, 1985. [4] H.W. Liiliefors, "Some confidence interval for queues", Oper. Res. 14, 723-727 (1966). [5] K.S. Trivedi, Probability and Statistics with Reliability, Queueing and Computer Science Applications, Prentice-Hall, Englewood Cliffs, NJ, 1982.

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