A NOTE ON THE JOINT DISTRIBUTION OF THE DURATION OF A BUSY PERIOD AND THE TOTAL QUEUEING TIME* Wa1ter L. Smith

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1 A NOTE ON THE JOINT DISTRIBUTION OF THE DURATION OF A BUSY PERIOD AND THE TOTAL QUEUEING TIME* by Wa1ter L. Smith Depa1'tment o.f Statistics University of North CaroZina at ChapeZ Hill Institute of Statistics r4i~eo Series No June~ 1976 * Research supported by the Office of Naval Research under Grant No. N C-0550.

2 A NOTE ON THE JOINT DISTRIBUTION OF THE DURATION OF A BUSY PERIOD AND THE TOTAL QUEUEING TIME.* by Walter L. Smith University of North Carolina 1. IntrOduction Co~ter simulation of queueing models is becoming the parammmt method for their investigation; complexities of the model which present fonnidable difficulties to any analytical attack can often be dealt with in a trivial way in the computer program. Nonetheless the existence of exact mathematical formulae is useful, even when they refer to rather specific llddels. For one thing they can indicate t in a rough'. way t the effect of varying relevant parameters. For another thing, they can provide valuable checks on the correctness of a computer simulation for \'lhich, maybe, a subtle tmdetected flow in the program is yielding results which are wrong, but not so conspiciously so as to arouse suspicions. For a further thing, the exact fonnulae can possibly provide indication of the sanpling errors inherent in a proposed MJnte Carlo study of a queueing system. The present note is part of an attack on a number of analytical problems which arose from work being done by Andrew Seila of the Curriculum of Operations Research at the University of North Carolina, Chapel Hill. * Research supported by the Office of Naval Research tmder Grant No. NOOOI4-76-C-0550.

3 2 He is investigating1 mainly by computer methods 1 the estimation of quantiles of the waiting time distribution in various queues; his methods focus on the busy period as the basic sampling mit. One check on the correct operation of his methods is through a study of the joint distribution of the total time lost (2) by customers in queueing during a busy period and the length (U) of that period. This note tackles this problem for the M/r'1/l queue and shows that product moments of 2 and U are obtainable in analytic form. They are given later in this note up to order three. 2 Derivation of a Joint Generating Function Let a busy period begin by a customer Co ' say, with service-time X1 arriving to find the server free. Let U be the length of the busy period thus L'rlitiated and let Z be the stun of the queueing-times of all customers who are served in that busy period. Plainly the joint distribution of U and Z depends upon x. Let a > 0 and 13 > 0 be dur:1il1y transform-variables and set (2.1) M(a 1 13Ix) = E{e- az - 13U I x}, where the conditional expectation has the obvious meaning. For ease we shall write M(a 113Ix) simply as M(x). The distribution of the ti.rne from the arrival of Co to the next customer C l ' say1 has a p.d.f. of exponential form and this leads us to the integral equation (2.2) M(x) = e- Ax - 13x + f: Ae- AU -(3u-a(x-u) J: M(x+z-u)~e-~zdzdu. In this equation the constants A and ~ are the intensities of arrival and service, respectively.

4 3 If we provisionally set then a straightforward computation yields the fact that the Laplace transfonn~ denoted by the notation: and defined for real (2.3) s > 0, is given by o 0 oo(s) = M (~)-M ~ s+a-~ (s+~) If this result is used in (2.2) we obtain after a little computation (2.4) rrp( ) = {t,p (~) +1 (s+aj}. s S+A+S s+a+13 s+a-~. Since we are principally concerned with the unconditional expectation E{e- az - SU } =J: ~~(x)e-~ dx = w.p(~) ~ it follows that MO(~) is of special interest to us. From (2.4) we obtain by putting s = ~~ and hence (2.5) ~bre generally, if n is any positive integer, then putting s = ~ + na

5 4 in (2.4) yields (2.6) MO(~+na) = ~+A+~+na {I + a(~~l) [MO(~) - MO(~+n+la)]}. Let us set ~ = A~/a, and (2.7) K(~) = 1 ~ ~2 - Ai~~8 + 2(A+~+S)(A+v+B+a) ~2 + etc. It is not difficult to see that this infinite series is always convergent and that K(~) is actually an entire fu"1ction. Somewhat tedious and repetitive use of (2.6) in (2.5) will then show that (2.8) In obtaining this result it is helpful to note that I'P (na) -+ 0 as n -+ co. It is possible to express (2.7) as a certain Bessel function, and, indeed, if we temporarily set then (2.9) This result does not seem particularly tractable. The most we can hope

6 e to achieve is an expansion of (2.8) as a double Taylor expansion in powers of ex and 13 from which the joint moments (unconditional) of Z and U 5 can be extracted. 3 Expansion of Joint Generating Func~ion From a wel1-1mown property of the Bessel function (Watson~ 1958) we have that ~ for any argument y ~ say, from which one can derive J () J () = 2(a~1) JeeY). e-l y Y, and thus obtain a continued-fraction expansion of the ratio of Bessel functions in (2.9). we find that (3.1) Indeed if we now set a b = - ~ ~ 1 = :;----- a :;:.-..., (a+b) - la+2b)-etc. I 1 I I =a.: (a+b)- (a+2b)- (a+3b)-... Let us call this continued fraction C(a,b). Then~ evidently, (3.2) 1 C(a,b) = a - C(a+b,b).

7 e In particular we see 6 This quadratic equation gives us the equivocal result (3.4) C(a,O) = A+~+a ± I[(A-~)2+a2+2(A+~)a]. 2/fll 21(XiiJ However, the substitution Ct = and S = must make l-lf'ti 0 (~) = 1. Thus, from (2.9), we need = _A_+_~ ± _1-,(~A_-~=)_2_ 2/fii 2/fil and so we may conclude the plus sign to be correct, and hence C(a,O) = a+~ This result will be found to agree with the well-known formula (Cox and Smith, 1961) for the transfonn of the distribution of the duration of a busy period, for it yields from (2.9): (3.5) E(e-~) = (~~)~.{A~ +)(A+~~~)' - 4} = ~A {(A+~+B) + IrA+~+a)2-4A~. Let us adopt the nota-c.ion C for C(a,b) and ai + j. C (a,b) = C(a,b), 1J aa 1 abj and be prepared to contract C ij (a,b) even further to C ij. Then

8 7 (3.2) gives (3.6) C IO { -cz- = 1 - C 10 If we set y = yea) = C(a,O) and then (3.6) gives (a+b sb) -:z- =- C10(a+bsb) - COI(a+b,b) COl. C y.. =y.. (a) = C CasO), 1J 1J 1) (3.7) and (3.8) Y01 = - (..1:-). ~ y2_ l. Since the function y2j(yz_l) occurs frequently we shall denote it by cp. Thus (3.9) { YlO = _ 4> Z YOI - -4> If we return to (3.6) and perform further partial differentiations then we find 2 ze lo C Zo C C ZC C C (3.10) _ 11 = -C (a+b b) C3 CZ ZO ~ T = - CZO(a+bsb) Z ZC OI C ---r- -._- O2 = -CZO(a+b,b) C' C 2 Computation then yields the equations:

9 (3.11) It is plain that, Z 3 YZO = - ~ Y _ 4 Yll - 3" 4 Y _ 2 <b4 10 A,5 Y02 - ~. - ~ 'I' Y Y at the expense of greater and greater complication, one can continue to obtain equations of higher order. the following results: _ Y30 -:jf y Y21 = - y4 - y6 We merely list (3.12). 6 A, A,6 + 4 A, A,7 Y12 - '4."" 'I' ""4 'I' -0 'I' 6' 'I' Y Y Y Y _ 6 A,3 6 A, A,5 36 A,6 12 A, A,6 228 A,7 Y 'I' -"4 'I' ""4 'I' - '4."" 'I' - 6' 'I' 0" 'I' - -0 'I' y 4 y y y y y y Let y.. denote the value of y.. when we set S = o. Similarly for 1J 1J Y and (j). Then it is easy to see that y = (Vj1)~ and ep = :>,,/(A-j1). (Note that we assume 11 >:>.. for stability of the queue, so (j) < 0.) Then we have the expansion 8 and so (3.13) Wi\11) = (~) ~ C(a ~b) = C(a,b) y = {(1..)YIO + Y irdj We are now in a position to extract the requisite moments, for we see that

10 (3.14) ruiz j = (_l)i+j ;tj 'J (Al1)~ 1+J Y Thus the equations (3.9), (3.11), (3.12) will yield product moments up to those of the third order. and spare the reader the detailed calculations. We list our results in the next section, 9 4 The Product Moments The following results are obtained as outlined in the previous sections. (4.1) Ell = ~ h~p). (4.2) (4.3) (4.4) (4.5) EZ=l.(P)2 A FpJ. 2 Var U = 1 p (l+p). ~ (1_p) Var Z =1-. p (2+7p+p ) 1. 2 (1_p)5 Cov (U,Z) - 1 p3(3+p) - i2 (1_p)4 From the last three results we obtain for the correlation coefficient PU,z ' say, (4.6) (3+p){P" Thus, as p -)0 1, Pu,z + ~ = , and, as p + 0,

11 10 Puz...,L;p ~ IZ Actually~ the correlation of U and Z is high for looderately low values of p~ as this table displays: -- Traffic Intensity p Pu Z ~ (The arrows in the bottom line indicate limit values.) For product moments of order three we find:

12 11 References Cox~ D.R. and Smith, W.L. (1961), Queues, Methuen & Co. Ltd., London, England. Watson, G.N. (1958), A Treatise on the Theory of BesseZ Functions 3 2d. ed. Cambridge University Press, England.