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A APPROXIMATIONS IN FINITE CAPACITY /? MULTI-SERVER QUEUES WITH POISSON ARRIVALS

by SHIRLEV A. NOZAKI

and SHELDON M. ROSS

9' o CJ>

\

OPERATIONS RESEARCH

CENTER

ORC 77-34 DECEMBER 1977

to: v^:

78 09 05 \50

UNIVERSITY OF CALIFORNIA • BERKELEY

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APPROXIMATIONS IN IINITE CAPACITY

MULTI-SERVER QUEUES WITH POISSON ARRIVALS^

I \

Operations Research Center Research Report No. 77-34

Shirley A./Nozakl and Sheldon M./ROSS

)ec ;1 / - U

77 / Hi /

U. S. Army Research Office - Research Triangle Park

U5 Ü/7 71^-1 I'C 71% \/DAAG29-76-G-0042

Operations Research Center University of California, Berkeley

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REPORT DOCUMENTATION PAGE *l»2«' S.MHf * I OOVT ACCCIIION NO

A??RCXI>!AT10NS IN FINITE CAPACITY ha'LT I-SERVER QUEUES VITH POISSON ARRIVALS

Shirlev A. Nozaki and Sheldon M. Ross

l »CM^OMMINQ OKOANIIATION NAMI *».O «ooncti Operations Research Center University of California Berkeley, California 94720

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Also supported by the Office of Naval Research under Contract N00014-:7-C-0:99.

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Queuelng Multi Server Finite Capacity Average Delay Approximations

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ABSTRACT

In (his paper^-we^consider;an M/G/k queuelng model having finite capacity N . That Is, a model In which customers, arriving In accordance with a Polsson process having rate

■h > , enter the system If there are less than N others present when they arrive, and are then serviced by one of k servers, each of whom has service distribution G . Upon entering, a customer will either Immediately enter service If at least one server Is free or else join the queue If all servers are busy. Our results will be Independent of the order of service of those waiting In queue as long as It Is supposed that a server will never remain Idle If customers are waiting. To facilitate the analysis, however, we wiH-, suppose a service discipline of «"first come first to enter

service. **'

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APPROXIMATIONS IN FINITE CAPACITY MULTI-SERVER QUEUES WITH POISSON ARRIVALS

bv

Shlrlev A. Nozaki and Sheldon M. Ross

0. INTRODUCTION

In this paper, we consider an M/C/k queueing model having finite

capacity N . That is, a model in which customers, arriving in accordance

with a Poisson process having rate \ , enter the system if there are less

than N others present when they arrive, and are then serviced by one of

k servers, each of whom has service distribution G . Upon entering, a

customer will either immediately enter service if at least one server is

free or else join the queue if all server» are busy. Our results will be

Independent of the order of service of those waiting in queue as long as It

is supposed that a server will never remain idle if customers are waiting.

To facilitate the analysis, however, we will suppose a service discipline

of "first come first to enter service."

Our objective is to obtain an approximation for the average time spent

waiting in queue by an entering customer. This is done mainly by means of

an approximation assumption, presented in Section 2, and used in Section 3

to derive the approximation. In Section 4, we let N ■ * and relate the

approximation to the existing literature.

isaa:

rir. luBwa.TriTav . a..:

1. BASIC DEFINITIONS AND FUNDAMENTAL EQUATION

We shall need the following notation:

P : the steady state probability that there are 1 people in the

system.

S : a service time random variable, I.e., P{S j_ x) - G(x) .

W : the average amount of time that an entering customer spends

waiting in queue (does not Include service time).

L. : the (time) average number of customers waiting in queue.

V : the (time) average amount of work in the system, where the work

in Che system at any time is defined to be the total (of all

servers) amount of service time necessary to empty the system of

all those presently either being served or waiting in queue.

V : the average amount of work as seen by an entering arrival.

We will make use of the following idea (previously exploited in such

papers as [1], [2] and [8]) that if a (possibly fictitlonal) cost structure

is imposed, so that entering customers are forced to pay money (according

to some rule) to the system, then the following identity holds—namely,

time average rate at which the system earns - average arrival rate

(1) of entering customers * average amount paid by an entering customer.

A heuristic proof of the above is that both sides of (1) times T is

approximately equal to the total amount of money paid to the system by time

T , and the result follows by dividing by T and then letting T -► « .

f A rigorous proof along these lines can easily be established in the models

we consider since all have regeneration points. More general conditions under which it is true are presented in (1).

■ • .—..

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Pv choosing appropriate cost rules, many useful formulae can be

obtalnfu us special cases of (1). For Instance, by supposing that each

customer pays $1 per unit time while In service. Equation (1) yields that

average number In service "Ml - PN)E(S] .

Similarly, by supposing that each customer pays $1 per unit time while

waiting In queue, we obtain from (1) that

LQ " V(1 " VWQ •

Also, If we suppose that each customer In the system pays $x per unit time

whenever Its remaining service times id x , then (1) yields that

(2) \(1 - PN)E SWQ + J (S - x)dx \(1 - PN)[E(S)WQ + ElS2]/2]

where W is a random variable representing the (limiting) amount of time

chat the n entering customer spends waiting in queue.

Another important fact which we shall use is that, since our arrival

stream of customers is a Foisson process, the probability structure of what

an arrival observes is identical to the steady state probability structure

of the system.

1

ii i «"-uww^ H. JJvi««»j..jrjWs,_i^-<iyr.rv.— ^'[nLWA'~Jm\f l "f ^J.'.!-L '^ ' - ■" J

IHK. .MM'KOXIMAIION ASSl'MniON

Lot C Jtnu>te tlu* injut I Unlvim vl 1st r ilnit IIM\ v'l C. , ['hat Is,

also K-t

■ ( X ) • I * ' ■ M -' •< .1 v

1 it x - v (x,y) -

lo It x »» v

We assume throughout that JxdG (x) - ElS'l/JE(Sl is finite. We make the

following approximation assumption.

Approximation Assumption:

Uiven that a customer arrives to find i busy servers, i s 0 , then

at the time that he enters service, the remaining service times of the

other i - Mi.k) customers being served has a Joint distribution that Is

approximately that of independent random variables each having distribution

G . e

Heuristic Remarks Concerning the A.A.:

1. In the infinite capacity case, the A.A. appears to be approximatelv

true either In heavy traffic Uhat Is, as \E[S) ■* k) or in light

traffic (.that Is, as \E[Sl • 0) . This Is so In heavy traffic

since the great majority of arrivals will encounter a large queue

and as a result the k departure processes tone for each serverl

they observe will be approximately Independent de laved renewal pro-

- -- ■ > ■ . ■ -

i. 'w» mi/..»-'.,ifi..i •'■ij.Jjipjui..'--_%.■;.„ ■BWRwmoHR ass.» r •JW^' .'^—z^r

cesses. Hence, considering those customers served by server 1 ,

It follows that when they enter service they would have been

observing k - 1 independent delayed renewal processes for a

large time, and the A.A. follows since the limiting distribution

of excess in a renewal process is just G .

In extremely light traffic, the great majority of arrivals

will find either 0 or 1 busy servers. Now, since Poisson

arrivals see the system as it is (averaged over all time), it

follows that arrivals finding 1 server busy would encounter the

same additional service time (for the busy server) as would random

(and uniform) time sampling of the excess of a renewal process.

Hence, the A.A. follows in light traffic from the renewal process

(excess) result.

Additional heuristics for the A.A. follows from the fact that it

is known to be (exactly) true when no queue is allowed (see [9]).

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3. THE APPROXUUTION

Since V Is the average amount of work as seen by an arrival, It

follows by conditioning upon whether or not an arrival enters the system

that

V ■ (1 - P.,)V + P., x (average work as seen by a lost customer). N e N

[

In accordance with our basic A.A., it sorms reasonable to approximate the

2 £[S 1

average work as seen by a lost customer by k ögTsi + (N - k)E(S] . Hence,

we have that

(3) (1 - PN)Ve + PMlk 4] E[S 2E(S + (N - k)E(Slj

Now for any arbitrary customer that enters the system we have the following

identity

work as seen by the entering customer *

k * time entering customer spends waiting in queue + R

where R is defined to be the sum of the remaining service times of those

being served when the customer enters service. Taking expectations yields

that

(4) V - kVL + E(R] . e Q

To obtain E[R] , we condition on B , the number of servers that are

busy when the customer enters the system:

■M ■—" i

IW"M 'vmA.m i.Ti i. 'gn^^igT; '-aL"^L^^T yi.!"^|gg^l..|wL"JJ!wy.^^ wi-.yi^.'.TTMir'Jii!.' ''-»j. <- '^ \ , n

(M

KIK) - EUIR B 1)

ElB - KB ,k)l |i|4 bv the A.A. e -11 >

Now.

(t>)

\(l - P )E[Sl ■ average number of busv servers as seen bv an arrival

- (1 - P..)EIBJ + kP . N e N

Also,

17) :U(Be.k)l - (l- PN- J/j)/^1 " V

and so from (3)-(7) and Equation (2) we obtain that

(3) U a l ' _ _ \ ''■■■ ' ' '

^N - k)PNE(S]

ei - pvuk - \E[sn

Therefore, it remains to obtain P and P , 0 ^ J ^ k - 1 . To do

so, we impose the following tictitional cost structure—namely, that the i

oldest customers in the system pay $1 per unit time, I ■ 1,^, .... k ,

where the age of a customer is measured from the moment it enters the

system. Hence, letting S ,S^, .... S. . denote k- 1 independent

random variables each having distribution C , we obtain from Equation (1^

that

VT-TTT CESCnaTr

P1 + 2P2 ••■ ... + (i - 1)P1_1 + 1(1 - P0 - ... - P1_1)

- X(P0 + ... + P^EIS) + ^P^MS - min (s^.S* S*)) J

(s - 2nd smallest of (s*. .... S*+1)) I + XPi+lE

(9) .

+ \P, _2Ens - (k - 1 - i)th smallest of (s*. .... S*_2)) 1

_2)EI(S - (k - i)th smallest of (sj, .... S*_1)) 1 + X(I - PN - P0 - ... - Pk

1-1, ..., k - 1

Pj + 2P2 + ... + (k - l)Pk.1 + k(l - P0 - ... - Pk_l) - X(l - PN)E[S1

(where x " { 1 • (0 if x < 0/

To» understand the above equations, suppose

first that i < k . Now, as only the i oldest pay, it follows that when

J customers are present the system earns at a rate j when J <. i end at

a rate i when j > i . Hence, the left side of Equation (9) represents

the average rate at which the system earns. On the other hand, an arrival

finding fewer than i customers already In the system will immediately go

into service and will pay a total amount equal to his service time; while

an arrival finding j present, k - 1 >. j >. i will also go immediately

into service but will only begin paying when j - i + 1 of the j others

in service leave. Thus, in this latter case, under the A.A., the arrival

would expect to pay a total of EMS - (j + 1 - i)th smallest of

(s^S®, ..., Se,\\ . Finally, if the arrival found more than k - 2 busy

^^***mmmmmmmm iz »A--AT-11.,.'w.iiuT'Ä.twü 4r,",fl'j ■ML # K^k'.KrT»:;-

Ht-rvers, thou hv will bt'uln puying at tor k - I ol tin»«»' ciistumors In

aorvlc'o whon ho entern HorvUo loavo the svKtora. Tills explains the first

k - I of the set of I'.tjuat lou il>). The last equation (when 1 - k) easily

follows since in this caso each oustomor will pay a total oqual to his tlrae

in sorvioe.

To simplify the set of Equation (*M , wo will need the following lonuna.

l.emna 1;

O ft* If S,S., ..., S are indepcnilent random variables such that S has

distribution G and the others Ü , then e

■[< El (S - 1th smallest of K ^f] ■ Vrr1 Eis'•

Proof:

Using the identity (x - y> - x - min ix.y') , we have that

{(s

S] - E min U ,

E IS - Jth smallest of S

E(S1 - E min (S , Jth smallest of S

1 *TI J

n K)] Now,

E min U , Jth smallest of S*. .... Se)

j PiS > alpljth snuillest of Ue S^) • njda

- f U - i'.(a)) "v lr\c, (u))l(l - G U))r"1da J Cm0\if o

■» '» H >I.IJ. ^i i»l'W'' f., .j^y,, .HBUttl.*'J*|..l«.«..v '»V'Bjir JKTnm'ZTrrrwir wr" TKIWaWL-Taz. ||

10

EIS Swj- (I - y) dv

.1-1

■^XW^HVtH1

EIS)

i-0

r ••• 1

which proves the lemma.■

It follows from Lemma 1 that the equations for P., and P ,

0 ^ J ^ k - 1 depend on C only through E(Sl . Hence, as the equations

are exactly true when G Is exponential, it follows (since for fixed V ,

It can be shown that the set of equations has at most one solution) that

the P. ,0^J^_k-l have the same relationship to P as when C. is

exponential. Thus, we are left to determine P , which we will approximate

by the answer in the exponential case. In other words, we shall use the

exact result for P, iO^.Jlk-l,P when C is exponential as our

approximation. This yields, from Equation (8), that

(ICH

ELsil V ^E[Sl)j ,N . k) E[Sl^E[SnN

2E(S) ~ . ,J Vk klk -k klk N-k

V liEiSlli .J-0 ^

Y 0E[Sl)j

J-k klk j-k (k - \E(S1)

U-LV ! • "■.,. '."'ILi.J''" iW.',WT *" '.' il mm—i

n

A. TUK INKINITK t'At'.UMTY CASK

In the lntlnlte c«p<icity CUH«? N - •■ , thf approxlnuil Ion (10^ reduce«,

when U:lSl k , to

un v^ - k. J k-1

:(k - UUk - \KlSl)' simDl (k - n:(k - \Kisi)

Some remarks are in order:

1. In [A], Klngman obtained bounds on W for the general queueing

system Cil/C/k. Wlten adapted Co the M/C/k case of Polsson arrivals,

his inequalities are

ELS'l [E[S'l + kAJ - (E[Sir/kl 2E(Sl(k - \E(Sl) " 2EIS1

wg - 2(k - \ElS))

It is easily verified that our approximation for VK is con-

sistent with Kingman's upper and lower bounds.

2. In (5), Klngman conjectured a heavy traffic approximation for W

in GI/G/k models. In the special case of Poisson arrivals, his

conlecture is that

w „ \-Eis-) - rjEmr t !L when xm] * k

Calling the right side of the above K and our approximation, as

given by (8^, N - R , we have

a

w^^^mm Mi*i iRiiifi \- —gw "i»i'.»jL^uiy: nj&'iT I'ii .jtjMgi.- -,

12

_-K_ . ÜLÜ + EIS] (k2 - (AE[S1)2\ N-R kP. P.EIS2^ >k /

k-1 where P ■ 1 - I P. . Hence, since In heavy traffic,

k 0 J

E[S] % k/\ , P. « 1 ( we see chat our approximation is consistent

with Kingman's heavy traffic conjecture.

3. Numerical tables for L- have been published by Hillier and Lo

in the special case M/E /k, where E represents an Erlang

distribution with r phases. Table 1 compares our approximate

formula for LQ (- XW ) with the Hillier-Lo tables.

4. Another heavy traffic conjecture was given by Maaloe who in [6]

conjectured that for the model M/E /k

WÄ a XE[S21

Q 2k(k - XE[S]) when XE[Sl as k .

As the ratio between our approximation and the above approaches

unity in heavy traffic, we see that our approximation is also

consistent with this conjecture.

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REFERENCES

[I] Brumelle, S. L. , "On the Relation Botwefii Customer aiul Time Averages In Queues," Journal uf ApplieJ Probability. Vol. «, pp. 508-520, (1971). " "•"""" "- ' ' "

[2] Brumelle, S. L., "A Generalization of L - ^W to Moments of Queue Length and Waiting Times," Operations Research, Vol. 20, No. b, pp. 1127-1136, (December 1972).

[3] Hlllier, F. S. and F. D. Lo, "Tables for Multiple-Server Queuelng Systems Involving Erlang Distributions," Technical Report No. 31, Department of Operations Research, Stanford University, (December 1971).

(4) Kingman, J. F. C, "Inequalities in the Theory of Queues," Journal of the Royal Statistical Society, Series B, Vol. 32, pp. 102-110,

[51 Kingman, J. F. C., "The Heavy Traffic Approximation in the Theory of Queues," Proceedings of the Symposium on Congestion Theory, pp. 137-170, University of North Carolina, (1965).

[61 Maaloe, E., "Approximation Formulae for Estimation of Waiting-Time in Multiple-Channel Queuelng System," Management Science, Vol. 19, No. 6, pp. 703-710, (February 1973).

[71 Newell, C, "Approximate Stochastic Behavior of n-Server Service Systems with Large n ," Lecture Notes in Economics and Mathe- matical Systems, M. Beckmann, G. Goos and H. P. KUnzi, eds., Springer-Verlag, (1970).

[81 Stidham, S., "Static Decision Models for Queuelng Systems with Non- Linear Waiting Costs," Technical Report No. 9, Stanford University, (1968).

[91 Takics, L., "On Erlang's Formula," Annals of Mathematical Statistics. Vol. 40, No. 1, pp. 71-78, (1969).