AD-756 668

BOUNDS ON THE DELAY DISTRIBUTION IN GI/G/1 QUEUES

Sheldon M. Ross

California University

Prepared for:

Office of Naval Research Army Research Office-Durham

January 1973

DISTRIBUTED BY:

National Ticlinical Information Service U. S. DEPARTMENT OF COMMERCE 5285 Port Royal Road, Springfield Va. 22151

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BOUNDS ON THE DELAY DISTRIBUDON IN Gl/ G /1 QUEUES

lleproducod by

NATIONAL TECHNICAL INFORMATION SERVICE ~

U S D•portment of Commerce Springf;ofd VA 211"

COLLEGE. 0 ENGINIERI

NJVERSITY OF CALIF . -

BOUNDS ON THE DELAY DISTRIBUTION IN GI/G/1 QUEUES

by

Sheldon M. Ross Department of Industrial Engineering

and Operations Research University of California, Berkeley

JANUARY 1973 ORC 73-1

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This research has been partially supported by the U.S. Army Research Office-Durham under Contract DA-31-124-ARO-D-331 and the Office of Naval Research under Contract N00014-69-A-O200-1036 with the University of California. Reproduction In whole or in part is permitted for any purpose of the United States Government.

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University of California, Berkeley

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Unclassified lh. OPOUP

1 BFPORr TITLE

BOUNDS ON THE DELAY DISTRIBUTION IN GI/G/l QUEUES

4 DCSCRIPTIVE NOTEt (Typ* ol report /ind.lncluslv dalim)

Research Report 5 AuTHOniS) (Firs! name, middle initial, last nair»)

Sheldon M. Ross

6 REPORT DATE

January 1973 e«, CONTRACT OR GRANT NO.

DA-31-124-ARO-D-331 b. PROjeC T NO.

20014501B14C

7«. TOTAU NO. OF PASES lb. NO. OF REFS

M. ORIGINATOR'S REPORT NUMBERIS)

ORC 73-1

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10 DISTRIBUTION STATEMENT

This document has been approved for public release and sale; its distribution Is unlimited.

n SUPPLEMENTARY NOTE» Also supported by the Office of Naval Research under Contract N00014-69-A-0200-1036.

12. SPONSORING MILITARY ACTIVITY

U.S. Army Research Office-Durham Box CM, Duke Station Durham, North Carolina 27706

13 ABSTRACT

SEE ABSTRACT.

DD.FNr.s1473 <PACE1' S/N 0101-807-681 1

lb Unclassified Security Classification

A-3140a

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Unclassified Security Cliissifii-ation

K EV WORDS ROLE

Delay Distribution

GI/G/1 Queues

Martingale

Stopping Time

NBU

NWU

IFR

DFR

DD ;r..l473 'BACK) <<u N Jl,'! •.■»07-ti'<;i

Unclasslfled Security Classification

" ,'-l«**>.*!«»M*j.( A-3U09

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ABSTKACT

Bounds are obtained for the limiting distribution of the delay in queue for a GI/G/1 system via Martingale theory. These bounds are somewhat stronger than similar bounds recently obtained by Kingman. Simplifications of the bounds are obtained in the special cases where the ser- vice distribution is either 1FR, DFR, NEU or NWU.

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1. INTRODUCTION \

Consider the usual Gl/G/1 queue with interarrlval times between customers

Xj^.X , ... and «ervlce times ^.Y., ..., where ECY^ < ECX^ < " . Let

U. = Y. - X, and assume that there exists a nonzero value 6 such that i i i

[•"'] = 1 . If such a value of 6 exists then by Jensen's inequality It

mus t be positive since EU, < 0 . Let D denote the delay in queue of the

n customer, and let

D(t) - 11m P{D > t} . n~ n

In [1] and [2] the following inequality was proven by Kingman

ae"et 1 D(t) < e"8t

where

a - InffdFCy)/ /V(y-0<lP(y) 'T. t J

and where F is the distribution of U. . Kingman proved the right side of

the above inequality in [1] by using Kolmogorov's inequality for Martingales,

and used a different technique in [2l to obtain the complete inequality. Fol-

lowing the Martingale approach of Kingman but using an appropriate stopping time

rather than Kolmogorov's inequality a somewhat sharper inequality will now be

obtained.

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2. THE INEQUALITIES

As was shown by Llndley [3]

D(t) = P |!v t for some n ■ 1,2, ... } .

n e I "i

3 •L

If we let Z ■ e , then as Z Is the product of Independent random

variables each with mean 1, It follows that \Z nj>l} Is a Martingale. For

a fixed positive constant A , define the stopping time N. by

NA - 1st n such that either Z > e or Z < e . A n n

As Is well known the moment generating function of N. exists In a region

about 0 and thus by Martingale theory

1 - E(Z ) - E n W

(1) - E

9K NA 1 X (p. >)

N, 9fül NA

• 1 II "i < " A < - A

m Now, NA

f (2) E

e lu e 1 'l

NA

1 1 • ■ ; *hl

e8tE (l^A

By conditioning on N. and ^ U. we obtain that A 1 i

l\>t

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■ 5

(3) inf E 0<r

■e(u-r) e 1 1 "i > rJ i

(^1-) N A

1 £ sup E e " I Ul * r

0<r L

Also, since

lim P A-x» (pi'') P O U > t for some n < »j

ll 1 D(t)

and m •

NA elu1 NA

lim E e 1 1 ,? Ui < ' A

A-»« 1 J

we obtain from (1). (2), and (3). by letting A * - that

■ V -

, i

-et -et

(4) sup E 0<r

"e(Vr> , '" * I U1 > r

< D(t) <

inf E|e 0<r

5 6

eOJj-r) | U, > r_

Note that the left side inequality is just the left side of Kingman's inequality

while the right sided inequality is stronger than Kingman's.

A somewhat weaker though probably more useful inequality based on the

service distribution can be obtained from the above as follows:

E fecvr) "I re[Y1.(x1+r)] . t' # e 1 | 1^ > rj - E|_e I Yl > xl + rJ •

Hence, by conditioning on Xj^ we obtain that

ffirv-^ 1 re(u,-r) fed.-f) e ' I Y, > sj< ELe ' 1 I Ü, > rJ < sup ELe | Y, > sj

inf E 0<8

VMM

The sup and inf are taken over all nonnegative values of r for which The sup

P^ > r) > 0 ,_- ~~^*fKrM,i*lm».4,

As this is true for all r , we obtain that

sup E 0<r

inf E 0<r

fedi -r) I re(Y-8) "1 [e •L | U1 > rj <. sup E [e x I ^ > sj

re(urr), i re(v8>, i Le I U^rJilnfE^ lY^sJ

(5)

Thus, from (4) we obtain

-et

sup E 0<s u

eCY^a) Yj^ > s

< 5(0 < -et

Inf E 0<s

• | Y1 > sj

A very Important special case occurs when the service times are exponentially

distributed with mean 1/y . That Is, when the system Is a G/M/l queue. In

this case, using the lack of memory of the exponential distribution we have that

the conditional distribution of Y. - s given that Y. > s Is Just exponential

with mean 1/y . Hence,

sup E s

feCY-s) 1 TeCY.-s) "I [e 1 I Y1 > sj - inf E[e 1 | yi > sj - -^

M - e

and thus In the G/M/l case

BCO-jfj.""

In certain special cases the sup and Inf In equation (5) can be more

easily expressed. We say that the service distribution G Is NBU If

G(s + t) <. G(s)G(t) for all s,t >. 0

and It Is said to be NWÜ If

The sup and Inf In equation (5) are taken over all nonnegative values of s for which POf. > s) > 0 .

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G(s + t) >. G(s)Q(t) for all s,t >. 0

where G(t) - 1 - G(t) . Thus G is NBU (NWU) means that the remaining

service time of a customer who has already been in service for some fixed

time is always stochastically smaller (larger) than the service time of a

customer just entering service.

Since V being stochastically larger than W implies that

E[f (V)] > E[f (W)] for all Increasing functions f , we obtain that if G

is NBU with 6(0) « 0 then

sup E 0<8

!>>-•> lYi>J4i while if G is NWU with G(0) - 0 then

inf E 0<8

re(Y-8) 1 feY/l

If we make the stronger (than NBU) assumption that G is IFR (that

is, that —* *• decreases in t for all s ) then it easily follows when G(t)

G(0) - 0 that

sup E 0<s

[e6^ | Yl > .] - .[.i and

inf E 0<8

re<Yr8) , 1 re(V8) , 1 [e A I Y. > aj - lim Ele • I Y. > si

S 'n

where M - sup{t : G(t) > 0} .

The terminology NBU (new better than used) originated in reliability litera- ture where it means that if G is the distribution of the lifetime of a component then the remaining life of any s year old (that is, any used) item is stochastically smaller than the lifetime of a new item.

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Similarly if G is assumed to be DFR (that is, if it is assumed that

—^ *■ increases in t for all s) then it follows when G(0) - 0 that G(t)

sup E 0<s

[9(^-8) . 1 feCY.-s) 1 [e x I Yj^ > sj = lim ELe A | Yj^ > sj

inf E 0<s

fea-s) 1 feY."] [e 1 | Y1 > sj - E[e ^ .

The importance of DFR service time distributions partly derives from the

fact that If the actual service distribution is a mixture of other distributions

each of which is DFR (for instance, each may be exponential) then the service

distribution is also DFR .

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REFERENCES

[1] Kingman, J. F. C, "A Martingale Inequality In the Theory of Queues," Proceedings Cambridge Philosophical Society. 59, 359-361 (1964).

[2] Kingman, J. F. C., "Inequalities in the Theory of Queues," Royal Sta- tistical Society Journal. Vol. 32, 102-110 (1970).

[3] Lindley, D. V., Proceedings Cambridge Philosophical Society. 48, 277-289 (1952).

[4] Marshall, K. T., "Some Inequalities in Queueing," Operations Research. 16, 651-665 (1968).

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