Waiting time distribution for the M/M/m queue

Waiting time distribution for the

M/M/m

queue

W:C. Chan and Y.-B. Lin

Abstract: A novel method is presented for the calculation of the waiting time distribution function for the M/M/in queue. It is shown that the conditional waiting time obeys an Erlang distribution

with rate inp, where p is the service rate of a server. An explicit closed form solution is obtained by means of the probability density function of the Erlang distribution. The derivation of the result proved to be very simple. The significance of Khintchine's method and its close relation to the proposed method is pointed out. It is also shown that the waiting time distribution can be obtained from Takacs's waiting time distribution for the C/M/in queuc as a special case. This reveals some insight into the significance of Takacs's more general, but rather complex, result.

1 Introduction

Consider a queueing system with an unlimited waiting room and m servers. Suppose that customers arrive at the queueing system at the instants T", 7I ,_,, 7

,,,...,

where by convention T" = 0, and for n = I , 2, 3,. . . , the interarrival times r!(+ I-7. are identically distributed, independent. positive random variables with distribution function

Assume that the system is work conserving, i.e. there is no idle server if there is a waiting customer. Customers are served in their order of arrival and the service times are identically distributed, independent random variables with the distribution function

which is independent of {T.}. This queueing system is known as the Erlang delay system which has been investigated intensively in the literature [1-5]. The main concern of the waiting time is its stochastic behaviour and the determination of its distribution function. Khintchine [2] presented a very thoughtful approach and obtained an analytical closed form expression for the waiting time distribution function. This paper presents

a simple

and novel alternative method for the determination of the waiting time distribution function. which gives new insight into the waiting time for the Erlang delay system. For a more general study of the G/C/m queue, the reader is referred to [&13]. Note that in order to study the properties of the associated queueing process, these C/C/m studies all resulted in having the problem of solving integral equations, and no analytic closed form solutions such as Takacs's for

the C/M/s queue were obtained.

~ ~ ~~~

0 IEE. 2003

IEE Prowedings online no. 20030274

doi: 10.1049/ip-cum:20030Z74

Paper fin1 received 30th August 2002 and in revised farm 7th January 2003 The iluthon are wirh rhe Drpanmem of Computer Sciencu .4 Information Engineeeng, National Chiao Tung Univrisity. 1001 Ta Hsaeh Roild. Taiwan

3iKIS0. Republic of China

IEE Proc. Coni,nm, Yo/, IJU, No. 3, June 2W3

2 Some fundamental results

Since customers are served in the order of their arrival, the investigation of the stochastic behaviour of the waiting time can be reduced to that of the state of the system. We shall summarise some well known results that are needed for the investigation of waiting time.

2.1 Stationary probabilities for states

of

the

M/M/m

queue

Denote by N ( f ) the total number of customers waiting or

being served in the system at the instant f. We say that the system is in state k at the instant f if N ( I ) = k. Further, let

Pk(r)

= Pr[N(t) = k ]

denote the probability that the system is in state k at the instant 1. If l i m p , then the limit

lim P k ( t ) = pk! for k = 0, 1 , 2 , r-m

always exists and is independent of the initial distribution

{Pk(0)},

k=O, 1, 2, ... If the limiting distribution {pa} exists, then it is uniquely determined by the following system of linear equations

and n,&) is the transition probability under the condition that the inter-arrival time is x. We shall quote the fundamental results for the limiting distribution

bk},

k = O , 1,2,

,.,,

as follows [2, 3, 41:

and

where U = %/p is the offered traffic or traffic intensity.

2.2 Erlang distribution

The Erlang distribution of order k with parameter 1~ has the distribution function

whose probability density function is

When k = 1. this probability density function reduces to the negative exponential density function. The Erlang random variable in ( 5 ) may be considered as a service time which is the sum of k independent random variables, each of which has an exponential distribution defined by (2). This service time was employed by Erlang in his method of stages. It is important to point out that the exponential distribution has the memoryless property. This means that the distribution of the remaining time, i.e. the residual time, for an exponentially distributed random variable is independent of the age of that random variable. This memoryless property plays an important role in the determination of the waiting time distribution function for M / M / m and G/M/ni queueing systems.

3

Let W denote the waiting time. I t would be more convenient and simpler to compute the probability Pr[W>/] than Pr[Wst], where Pr[W>t] denotes the probability that a test customer entering the system at a random moment has a waiting time greater than

r.

Furthennore, let Pr[W>tlN=k] be the probability of the event { W > l ) on the condition that on arrival the test customer finds the system in state k . Using the formula of total probability, we can write

The waiting time distribution function

where nk is the probability that the system is in state k just prior to the arrival instant of the test customer. ink}, for k=O, 1. 2, _.., is known as the arriving customer's distribution. Since the amval process is Poisson, the amving customer's distribution i n k } and the outside observer's distribution ipk] are equal. Then we have [3]

Z,i = pk> k = 0: 1 : 2 , ... (8)

Equation (X) implies the PASTA (Poisson arrivals see time averages) property.

Since Pr

w > t l N = k ] =

0 f o r k < m and t > O , (7) reduces to

m

~ r [ ~ > t ] = C p k P r [ ~ > ~ l ~ = k ] (9)

k=m

It remains

to

determine the conditional waiting probabilities Pr[W>tlN=k] for k t n t because p k is given by (3).

Observe that during the waiting period of the test customer all m servers must be busy. Any one of these m busy semers can contribute a service rate IC. The resultant service rate of the ni servers as a group is essentially mp. In other words, the whole group of ni busy servers acts like a single server with an exponential service time of rate ni)l. At this point, k-m

+

1 customers are waiting for service in the system, and each of them will take an exponential service time of mean limp. It follows from the memoryless property of the exponential distribution that the waiting time of the test

160

customer is composed of k-ni

+

1 exponential service times, each of which has an exponential distribution with rate mp. Thus we can write the conditional waiting time as

where R is the residual (exponential) service time of the group of m customers being served on the arrival of the test customer, and Wj is the waiting time of thejth customer in the queue. Clearly, the conditional waiting time W;+,+l can he regarded as an Erlang random variable with the following probability density function:

x

2

O:k 2 m

Then the conditional waiting time distribution is simply m

Substituting (3) and (12) into (9) results in the waiting time distribution function

which is the desired solution.

4 Remarks

The simplicity of the integration in (13) results from the infinite sum, which yields an exponential function. If the M/M/m queueing system has only a finite waiting room, then the upper limit of the summation becomes finite and no exponential function results. In this case, it would he simpler to use the distribution of ( 5 ) for Pr[W>tlN=k].

4.1 Khintchine's explanation

Khintchine [2] noticed that if k > m , there must he k-m customers waiting for service in front of the test customer in the queue. If the queue discipline is first-come, first-served,

then

the

arriving test

customer finding k-m customers

waiting in front of him will obtain servicc after the (k- m

+

I)th departure. Thus the conditional probability P r [ W > t l N = k ] is equal to the probability that during the time interval I after the arrival of the test customer, there will be at most k-m departures of customers. It follows that

whereL(i] denotes the probability of exactly j departures in the time interval I , which remains to be determined. Since the service times are exponential, the probability that no departures occur during the time interval f from the arrival IEE Proc. C ~ ~ m m w . , VU/. 150. No. 3, June 20013

and A is given by instant of the test customer is equal to

fo(t) = e-mi"

which implies that the inter-departure times have exponen- tial distribution with rate mp, and hence equivalently the number of departures during the time interval I obeys a Poisson process with rate fnp. Therefore, we hnve

,/;(t) =

py]

__

e-"'@', , j = 0 , l : 2 ; . ..

₍₁₅₎

This result indicates that when all

m

servers are busy in the time interval I , the process of departure follows a Poisson process with rate mp. Substituting (15) into (14) yields the conditional probability

which is the complementary distribution function of the Erlang distribution of order k-m+ 1 and parameter mp. Using (3) and (16) in (9), we find

(17)

Let i = k-mj. Then (17) can he rewritten as P r [ W > t ] =

(;ETm)

~

x

(cc(q)n7)i[(:)'y]}

m m

=

t

$

)

m

[

g

-

]

[i:(3']

i=o

j = O i-0

= ( ? ! E ~ ) ~ - ( m i 1 - ; + , m - a I

2

0 (18)

It is interesting to note that Khintchine's method essentially makes use of the complementary Erlang distribution function of (16) to calculate the conditional probability

Pr[W>tlN = k] = 1

-

E ~ - , , , + I ( I ) ₍₁₉₎

4.2

Takacs's

waiting time distribution for Palm input and exponential service times for the GlMlmqueue

Takacs investigated the multiple server queueing process for Palm input and exponential service times. He obtained a more general but very complex formula for the waiting time distribution [14]. We quote his formula as follows:

(20)

= N m d I - U ) ) (21)

A~-""'-''''

P r [ W < t ] = l - -

,

1 2 0 I - w

where w is the only root of the functional equation

in the unit circle, where

IEE Proc Cbmnmm.. Vol. 130. No. 3, Junp 2003

(23) and forj=O, I ,

2,

...

From (20), we deduce the following formula:

In principle, it is possible to obtain the waiting time distribution function (18) from (20) as special case when F(t) is defined by (1). However, this task is not trivial. It is instructive to show that this is indeed the case. In this case, we have found that

Substituting (25) into (23), we have

= p"'

From (25) and (26), then (24) can be rewritten as

as expected. 5 Conclusions

A novel method has been presented for the calculation of the waiting time distribution function Pr[W>t] for the M/M/ni queue. Although the result given by (13) is not new, the method of deriving the result is novel, and provides new insight into the conditional waiting time that has

a

complementary Erlang distribution of order k-m+ 1 and parameter mp. Also we have presented two alternative methods to obtain the same result. Khintchine's method is very logical and thoughtful, and offers a physical inter- pretation of the departure process of customers from a group of m busy servers. It is interesting to note that Khintchine's method implicitly makes use of the comple- mentary Erpang distribution defined in (19).

It has also been shown that given Takacs's formula for the C/M/m queue in (20), the same result (18) can be obtained as a special case. Since (20) is very complex, it appears that derivation of (13) from (20)

is

not a trivial task. To our knowledge, this task has not been carried out and hence is not available in the literature in the way presented in this paper. Also, this approach provides a much simpler conditional waiting time expression for use in performance modelling of wireless telephone networks [15].

6 Acknowledgments

The authors would like to thank the anonymous reviewers. Their cominents have significantly improved the quality of this paper. This work was sponsored in part by MOE Program for Promoting Academic Excellence of Univer- sities under the grant number 89-E-FA04.1-4, IIS, Acade- mia Sinica, FarEastone, the

Lee

and MTI Center for Networking Research, NCTU, and National Science Council under contract NSC 90-2213-E-009-156.

7 References I

2

MEDHI. J.: 'Stochastic models in queueing theory' (Academic Press, 1991)

KHINTCHINE. A.Y.: 'Mathematical methods in the theory of queueing' (Hafnner. New York, 1969. 2nd rdn.), (English translation

from Russian)

COOPER. R.B.: 'Introduction 10 queueing theory' (Elwvier Science.

New York, 1981, 2nd edn.)

KLEINROCK. L.: 'Queueing systems: Volume I-Theory'. (Wilcy, Ncw York, 1976)

3

4

5

6

GROSS, D.. and HARRIS. C. H.: 'FundnmenIals ofqueueing theory' (John Wiley. 1998)

KENDALL. D.: 'Stochastic processes m u r i n g in the theory of queues and their analysis by the method of imbedded Markov chains'. Ann. marl^. Slur.. 1954. 21. pp. 338-354

7 KIEFER. 1.. and WOLFOWITZ. J.: 'On the theory of queues with many S~NCTS', 7riln.5. Am. Muh. Soc.. 1955, 78. pp. 1-18

X fC4RLIN, S.. and McGREGOR. J.: 'The differential e q ~ t i o n s of birth-and-death processcs and the Stieltjes moment problem', Trurrs A m Marh. Soc. 1957. 85, pp. 489-546

9 KARLIN. S.. and McGREGOR, J.: 'Thc classification of hinh-and- death pr-scs', 7 r u m Am. Marlr Soc., 1957. 86. pp. 36W00 10 KARLIN, S.. and McCREGOR. J.: 'Many server queueing processes

with Poisson input and exponential ~ C N ~ C C times', POC. J. Morh.. 1958.

8, pp. 87-118

I I PRESMAN. E.: 'On the waiting time for many S ~ N ~ T queueing systems'. 7Iieog. Prubuh. Appl., 1965, 10. pp. 6 S 7 3

12 DE SMIT. J.H.A.: 'Some general results h r many s r r ~ e r qucues', Ad,!. Appl. Prohob.. 1973. 5. pp. 153-169

13 DE,SMIT. J.H.A.: 'On the many server queuc with exponential

X N ~ C ~ timcs'. Ado. Appl. Prohob.. 1973. 5. pp. 170-182

14 TAKACS, L.: 'Introduction

_{,_,>}

to the theory of qucucs' (Greenwood , ....,.,.

""',

I S TSAI, H.M., and LIN, Y.-B.: 'Modeling wireless local loop with general call holding l i m a and finite number of subscnben'. IEEE Trmr Corrzput.. 2002, 51. (7). pp. 77S7-786