The stochastic law of the busy period for a single server queue with Poisson input

JOURNAL

OF

MATHEMATICAL

ANALYSIS

AND

6, 33-42

APPLICATIONS

(1963)

The Stochastic Law of the Busy Period for a Single Server Queue with Poisson Input LAJOS Columbia

University,

Submitted

I.

TAKACS New

by Richard

York,

New

York

Bellman

IN-~RODUCTI~N

An interesting problem in the theory of queues is to find the stochastic law of the busy period for a single server queue. Suppose that customers arrive at a counter at times 7r, TV, ***, T,, .*a where the interarrival times 7, (?I = 1, 2, **.) are identically distributed, mutually independent Tn+1 random variables with distribution function P{Tn+1 - 7, 5 x} = F(x)

(n = 1, 2, e.0).

The input is said to be a recurrent process. The customers are served by a single server. The server is idle if and only if there is no customer in the system. Denote by xn the service time of the nth arriving customer, It is supposed that {xn} is a sequence of identically distributed, mutually independent, positive random variables with distribution function (?I = 1, 2, -)

P{xn 5 xl = Wx)

and independent of the input process. The busy period is defined as the time interval during which the server is continuously busy. Busy periods and idle periods alternate. Evidently every busy period independently of the others has the same stochastic law. Denote by G,*(x) the probability that a busy period consists of n services and has length 5 x. Define dG,*(x)

(1)

qs, w) = 2 T,*(s)w” n=1

(2)

T,*(S) = j,”

e+

for s(s) >= 0 and

for%(s)~OandIwlIl. 33 3

Evidently G,*(a) services and

is the probability

that a busy period consists

of n

G(x) = 2 G,*(x) n-i is the probability

that a busy period has length s x. Let

for m(r) 2 0. Several authors investigated the stochastic law of the busy period for different queueing processes. In 1942 Bore1 [1] found G,*(x) for Poisson input and constant service times. In 1951 Kendall [2] derived a functional equation for r(s) in the case of Poisson input and general service times. In 1952 Pollacxek [3] gave a complex integral expression for r(s, w) in the case of recurrent input and general service times. During the last ten years several methods have been used to investigate the stochastic law of the busy period for different queueing processes. The method of functional equations was used by the author [4,5]. RouchC’s theorem was used by Gaver [a], and the author [7-9, 51. The technique of difference equations was introduced by Conolly [lo, II]. Mathematical induction was applied by Prabhu [12]. Combinatorial methods were used by Tanner [13] and the author [14-17J. The method of integral equations was introduced by Rice [18]. Other problems concerning busy periods were investigated by Gani [19], Gani and Prabhu [20], Gani and Pyke [21], Karlin and McGregor [22], Karlin er al. [23], McMillan and Riordan [24], Riordan [25], and Tanner [26]. The aim of this paper is to find the stochastic law of the busy period for a single server queue with Poisson input and general service times. The results are more complete than the earlier ones in the respect that we find not only the explicit form of G,*(z) but the probability that the busy period has finite length or consists of a finite number of services.

II. AN AUXILIARY THEORETH Let H(x) be the distribution function of a nonnegative random variable. The trivial case H(x) = 1 if x 2 0, H(z) = 0 if x < 0 is excluded. Define

for X(s) 2 0. The Laplace-Stieltjes

transform

$(s) is a regular function of J

STOCHASTIC

LAW

FOR

SINGLE

SRRVRR

35

QUJNR

in the domain YQ) > 0 and is a continuous function of s in the domain S(s) L 0. Always 1#(r(s) 1 4 1 if S(s) 2 0, \ #(s) 1 < 1 if 9ys) > 0 and #(O) = 1. Let m a= x dH(x). (6) I0 If a is finite then a = - #‘(+ 0). Now we shall prove the following LEMMA.

If~>o,~(s)rOandIWI1.lthen2=y(s,W),t~rootOfthe

equation 2 = WI& + X(1 - 2))

(7)

that has the smallest absolute value, is V(S, W) = 2

7

s,” e-(a+s)z X+’

dH,(X)

where H,(x) denotesthe n-th iterated convohkon of H(x) with itself. Alwuys 1y(s, w) I 5 1. If Aa 5 1 then y(0, 1) = 1 and if Aa > 1 ihen y(0, 1) is real andOOand~w)<1. hOOF.

Define

for R(s) 2 0 and I w I s 1. We shall prove that y*(s, w) is a continuous functionofsandwif%(s) ZOandI wl $ 1 anditisaregularfunctionofsandw if S(s) > 0 and I w ( < 1. It is sufficient to prove that the series(9) is uniformly convergent in the domain S(s) 2 0, 1w I g 1 becausethe tih term of the series (9) is continuous in s and w if S(s) 1 0, I w I j 1 and regular if %(s)>Oand~w~
( $ e-b (xr)“-l

5 e-(*-l)) (n - I)“-l

(n = 1,2, -*-)

whence e-(1+8)=x+1 dH&)

I

5

e-w-l)(n

-

l)"-1

?l!

and the uniform convergenceof (9) follows from Euler’s formula

36

TAKkS

We note that (7) has at most one root on the periphery of the unit circle; namely, .a = 1 is a root if w+(s) = 1. If z # 1 and 1z 1 = 1, then 1z&s + A(1 - z)) ( < 1 and, therefore, z cannot satisfy (7). 1. Let hcl > 1. Then 1#(s + A(1 - z)) 1 < #(At) < 1 - E if E > 0 is small enough. For gL(he) and 1 - E are equal at E = 0 and their right hand derivatives at E = 0 are - ha: and - 1 respectively where - Aor < - 1. Consequently, 1w+(s + A(1 - z)) 1 < 1z ( if 1z 1 = 1 - E and E > 0 is small enough. Since both w#(s + A(1 - x)) and z are regular functions of z in the domain / z 1 2 1 - c, it follows by RouchC’s theorem that (7) has one and only one root, z = y(s, w), in the domain ( z I 5 1 - 6. The explicit form of y(s, w) can be obtained by Lagrange’s expansion. (Cf. [27, p. 1321.) Thus we get that y(s, w) = y*(s, w) if s(s) 2 0 and 1w 1 5 1. Here ~(0, 1) is a positive real number and 1~(0, 1) / < 1. This proves the theorem for hei> 1. 2. Let AN 2 1. First we prove that (7) cannot have two distinct roots in the unitcircleIzI~l.If%(s)~OandjzI~l,thenwehave

and

w + X(1- 4) zzz ( & ax

for

x = 1.

If we suppose that z1 and z2 are two distinct roots of (7) in the unit circle ) z 1 5 1, then we obtain that I 3 - Xl I = I w I I $Q + v

=lwll.i This contradiction

- x2)) - w

+ Xl - 4)

I

%a$@+ A(1- x)) dz
two cases:

(4 I wW I < 1. (Th is contains the case X(s) > 0.) If /z I 2 1, then I w#(s + A(1 - z)) ) 5 1w+(s) I < 1 - E for a sufficiently small f > 0. Consequently, j w&s + A(1 - z)) / < / z / if j z j = 1 - Q and E > 0 is small enough. Thus by RouchC’s theorem we can conclude that (7) has one only one root in the circle ( z ( 5 1 - E and the explicit form of z = y(s, w) can be obtained by Lagrange’s expansion. This proves that y(s, w) = y*(s, w) if 1w$(s) [ < 1 (therefore, if g(s) > 0). (b) I w#(4 I = 1. (Th en s(s) = 0 and I w I = 1.) Let lirnndoo s, = s where s(s) = 0 and %(s,J > 0. We shall prove that I y*cs, w) I 5 1

and

x = y*(s, w)

STOCHASTIC

is a root of (7). y(s,, w) = y*(sn, domain ‘X(s) 2 j y*(s, w) 1 2 1.

LAW

FOR

SINGLE

SERVER

37

QUEUE

Thus r(s, w) = y*( s, w) must hold also if s(s) = 0. Since w), 1y*(sn, w) 1 5 1 and y*(s, w) is continuous in the 0, 1w \ 5 1, we have lim,,, As,,, w) = y*(s, w) with Further, if n -+ 00 in r(sn, w) = w#(s + A(1 - r(sn, w)),

then we get Y”(S, w) = w$qs + A(1 - y*(s, w)). Since (7) has at most one root in the domain 1z ) 5 1, y(s, w) = y*(s, W) must hold also in the case %(s) = 0. Accordingly, y(s, W) = y*(s, w) if ha 5 1 and ‘R(s) = 0. If s = 0 and w = 1 then x = 1 is a root of (7) and, therefore, y(0, 1) = 1 if ha 5 1. This completes the proof of the lemma. REMARK 1. If ( w+(s) 1 = 1, then z = ~(s, w) is the root with absolute value in z of the equation

smallest

z = w+(s) #(A(1 - z)).

(11)

If(w#(s)I=l,then~w~=land~~(s)]=1.In

/ e+ 1 =( 1 and to satisfy ( #(s) 1 = 1 I‘t is necessary that the function H(X) be constant in every interval of x in which e-sr f #(s). In other words, H(x) can increase only at points x for which eesr = I&S). Hence, if ) #(s) j = 1, then qJ(s + A(1 - 2)) = 1,” e-[s+a(l-z)~r W(x)

= t)(s) #A(1 - 2)).

(12)

Putting (12) into (7) we obtain (11). REMARK 2. By forming I = 1,2, **awe obtain that

the

Lagrange

expansion

of

[As, w)]’

for

If we put s = 0 and w = 1 in (13) and suppose that Xa 2 1, then we obtain the following interesting identity (14)

38

TAIdcs

If, in particular,

in (14), then we obtain that

m1 (hU?l)“-’ Is n e-aan (?++=* n-r

(15)

for bar $ 1 and Y = 1, 2, *es.

III.

THE

hOBABILITY

LAW

OF THE

BUSY PERIOD

Now let us suppose that {T,,} is a Poisson process of density A, i.e., (16) whereas H(x) is arbitrary. G,*(x), is given by

Then C(S),

the Laplace-Stieltjes

transform

of

THEOREM 1. ff%(s)IOandIw/5l,then

3 T,*(s) w” = y(s,w) n-=1

(17)

where y(s, w) is the root with smallest absolute value in z of the equation 2 = w#(s + A(1 - 2)).

(18)

PROOF. Denote by G&.(x) the probability that a busy period consists of at least n services, that at the end of the nth service R customers are in the system and that the total service time of the first tt customers is g x. Then G,*(x) = G&(x). It is easy to see that

and for n = 1, 2, ***

-C!

LAW

FOR

SINGLE

SRRVRR

39

QURUE

Let (S(s) 2 0). F orming the Laplace-Stieltjes

Then I’,*(s) = I’&(s). (20), we obtain

(21)

transforms

of (19) and W)

and

If we introduce the generating function

(24) for 1I 1 $1

and 3(s) 2 0, then we have us,

2) = $(s + A(1 - 4)

and qk+&,

4 = $+ + q1

- 4) Wn(s,4 - cw

whence, for 1w 1 < 1, w$(s + A(1 - 2)) [a - 2 C%)

2 U*(s,2)

w”

=

2

W-1

- w+(s + X(1 “-l2),

w”]

*

(25)

The left-hand side of (25) is bounded if R(s) 2 0, 12 1 5 1 and 1w 1 < 1, because I Un(s,2) 1 i; 1. In this domain the denominator of the right-hand side of (25) has one and only one root 2 = r(s, w) defined by (8) and, therefore, this must be also a root of the numerator. Thus

2 F,*(s)ws = y(s, w)

(26)

n-1

if Yt(s) 2 0 and I w ( < 1. Since the left-hand side of (26) is a continuous function of w if I w ) 5 1 and R(s) 2 0, we obtain by continuity that (26) is also valid for I w I S; 1 and s(s) 10. This completes the proof of the theorem.

40

TAKkS

THEOREM2.

We have

G,*(x) = G

Ix e-A* un-’ dH,(u)

(n = 1, 2, **a).

0

(27)

PROOF. By (17) and (18)

I-,*(s)= 7,” *

,-~A+~,z~n-ldH,(~) 0

(28)

whence (27) follows immediately. REMARK3. The probability that in Y different busy periods altogether n customers are served and that these T busy periods have total length 5 3c is given by -r

n

2 I

0

e-aus

dH,(u).

For, the Laplace-Stieltjes transform of the probability in question is equal to the coefficient of w* in (13). THEOREM

3.

The probability that the length of a busy period is $ x is

given by G(x) = n$l s

*

jz e-AUunY1dH,(u). 0

Ij hci 5 1, then G(m) = 1, whereasij AU> 1, then G(a) = w < 1 where z = w is the only root of the equation z = I/!@(1 - z))

(31)

in the unit circle ) z 1 < 1. PROOF. If we add (27) for II = 1,2, e-e,then we obtain (30). The LaplaceStieltjes transform of G(x) is r(s) = y(s, l), i.e., the root with smallest absolute value in z of the equation

z = $(s + X(1 - z)).

(32)

Thus G(w) = r(O) = ~(0, 1). If ACY 5 1, then ~(0, 1) = 1, whereasif hor2 1, then 0 < ~(0, 1) < 1. Evidently, ~(0, 1) is the probability that a busy period has finite length or consistsof a finite number of services.

STOCHASTIC REMARK

4.

LAW FOR SINGLE

The rth iterated

SERVER QUEUE

convolution

of G(x) with (I

For, the Laplace-Stieltjes REMARK

5. c+ =

and 0, =

transform

of G,(x)

is [r(s)]’

itself is

= 1,2, e*.).

(33)

= [~(s, l)]‘.

Let ODx’dH(x)

= (-

l>‘$W(+

0)

(r = 1, 2, 7s.)

(34)

m x’dG(x) I 0

= (-

1)’ r”‘(

+ 0)

(r = 0, 1, *a.).

(35)

(Y = 0, 1, a**),

(36)

I 0

Since r(s) = y(s, 1) defined by (8), we obtain

0,=a% G Srn .

0

e-lx xn-lfr

that

dH,(x)

provided that it is finite. If we take into consideration that z = r(s) satisfies (32), then we can determine 0, (r = 1, 2, *..) recursively. If we suppose that hor, < I and c+ is finite, then O,, O,, .a*, 0, are also finite and 0, = 1,

(37) If ha, 2 I, then 8, (r = 0, 1, ***) can be obtained

in a similar

way.

REZERENCETS 1. BOREL, B. Sur l’emploi du theoreme de Bernoulli pour faciliter le calcul dun infinite de coefficients. Application au problbme de I’attente ti un guichet. Compt. rend. acad. sci. Paris 214, 452-456 (1942). 2. KENDALL, D. G. Some problems in the theory of queues. 3. Roy. Statist. Sot. Ser. B, 13, 151-185 (1951). 3. POLLACZEK, F. Sur la repartition des pdriodes d’occupation ininterrompue d’un guichet. Compt. rend. acad. sci. Paris 234, 2042-2044 (1952). 4. TAKACS, LAJOS. Investigation of waiting time problems by reduction to Markov processes. Acta Math. Acad. Sci. Hung. 6, 101-129 (1955). The transient behavior of a single server queueing process with a 5. TAI&S, LAJOS. Poisson input. Proc. Fourth Berkeley Symposium on Math. Statist. and Probability, Vol. 2, pp. 535-567. Univ. of California Press, Berkeley and Los Angeles, 1961. 6. GAVER, DONALD P. Imbedded Markov chain analysis of a waiting-line process in continuous time. Ann. Math. Statist. 30, 698-720 (1959).

7. T-cs, LAJOS. Transient behavior of single-server queuing prcsas~ with recurrent input and exponentially distributed service times. @crct&n &search 8, 231-245 (1960). 8. TA~~c% LAJOS. Transient behavior of single-server queueing processes with Erlang input. Trans. Am. Math. Sot. 100, l-28 (1961). 9. T&C& LAJJOS. The transient behavior of a single server queuing process with recurrent input and gamma service time. Ann. Math. Statist. 32, 12861298 (1961). 10. CONOLLY, B. W. The busy period in relation to the queueing process GI/M/l. Bbmetrika 46, 246-251 (1959). 11. CONOUY, B. W. The busy period in relation to the single-server queueing system with general independent arrivals and Erlangian service-time. J. Roy. Statist. Sot. Ser. B, 22, 89-96 (1960). 12. h.ABHU, N. U. Some results for the queue with Poisson arrivals. 3. Roy. Stutist. Sac. Ser. B, 22, 104-107 (1960). 13. TANNER, J. C. A derivation of the Bore1 distribution. Biometrika 48, 222-224 (1961). 14 TAK.~CS, LAJOS. The probability law of the busy period for two types of queuing processes. Operations Research 9, 402-407 (1961). 15. TAKACS, LAJOS. The time dependence of a single-server queue with Poisson input and general service times. Ann. Math. Stat&. (submitted). 16. Ttics, LAJOS. A combinatorial method in the theory of queues. 3. Sot. hi. Appl. Math. 10 (1962) to appear. 17. T~tics, LAJOS. A single-server queue with recurrent input and exponentially distributed service times. Operations Research 10, 395-399 (1962). 18. RICE, S. 0. Single-server systems--II. Busy periods. Bell System Tech. 3. 41, 279-310 (1962). 19. GANI, J. Elementary methods for an occupancy problem of storage. Math. Ann. 136, 454-465 (1958). 20. GANI, J. AND PRABHU, N. U. The time-dependent solution for a storage model with Poisson input. 3. Math. and Mech. 8, 653-663 (1959). 21. GANI, J. AND PYKE, R. The content of a dam as the supremum of an infinitely divisible process. 3, Math. and Mech. 9, 639-651 (1960). 22. KARLIN, SAMUEL AND MCGREGOR, JAM=. Many server queueing processes with Poisson input and exponential service times. Pacific 3. Moth. 8, 87-118 (1958). 23. KARLIN, S., MILLER, R. G. AND PRABHU, N. U. Note on a moving single server problem. Ann. Math. Statist. 30,243-246 (1959). 24. MCMILLAN, B. AND RIORDAN, J. A moving single server problem. Ann. Statist. 28,471-478 (1957). 25. RIORDAN, JOHN. Delays for last-come first-served service and the busy period. Bell System Tech. 3. 40, 785-793 (1961). 26, TANNER, J. C. A problem of interference between two queues. Biometrika 40, 58-69 (1953). 27. WHI~AKER, E. T. AND WATSON, G. N. “A Course of Modem Analysis.” Cambridge Univ. Press, 1952. 28, POLLACZEK, F$LIX. “Problemes Stochastiques Poses par le Phenom&ne de Formation d’une Queue d’Attente B un Guichet et par des Phenom&nes Apparentes.” Gauthier-Villars, Paris, 1957. 29. TAKES, LAJOS. “Introduction to the Theory of Queues.” Oxford Univ. Press, New York, 1962.