A note on a fluid queue driven by an M/M/1 queue

In this note we consider the fluid queue driven by an M / M /1 queue as analysed by Virtamo and Norros [5]. We show that the stationary buffer content in this model can be easily analysed by looking at embedded time points. This approach gives the stationary buffer content distribution in terms of the modified Bessel function of the first kind of order one. By using a suitable integral representation for this Bessel function we show that our results coincide with the ones of [5]. 1. A fluid queue driven by an alternating renewal process. Consider a ftuid queue with a constant leak rate Cl' The input of the ftuid queue is governed by an alternating renewal process XI,Yl ,X2,Y2, .... The sequences XI,X2, .•. , and Yt, Y2 , ••• , are Li.d. sequences with distribution function Fx(') and Fy('), respectively. During the alternating periods oflengths Xi, resp. Yi, the input rate ofthe ftuid queue is equal to C2 (> Cl), resp. Co « Cl). Although the results ofthis section can be easily extended to general values of Co, Cl and C2, we assume in the sequel for convenience that Co = 0, Cl = 1 and C2 = 2. Clearly as stability condition for the queue we have EX < EY. Our goal is to find the stationary buffer content distribution. Define Zi as the buffer content at the beginning of the i-th period Xi. Then the Zi'S satisfy the recurrence relation (with Zl := 0) (1) This relation is equal to the one for the waiting time in a GIG 11 queue with interarrival time distribution Fy(·) and service time distribution Fx(')' Hence we conclude that the distribution of the stationary buffer content, Z, at the beginning of an X-period is equal to the stationary waiting time distribution in a GIGll queue. If we introduce the random variable T as the stationary buffer content at an arbitrary point in time, then standard renewal theory arguments show that (see also Kella and Whitt [3]) (2) Z+X, w.p. EXI(EX+EY), max(Z + X f,O), w.p. EYI(EX + EY), where all random variables involved are independent and X (resp. f) denotes the residual life time of the random variable X (resp. Y). In the special case that the Yi '8 are exponentially distributed with parameter A d we have Y = Y and so we obtain from (1) and (2) that where d T = Z + X . l[U=l] u _ { 1, w.p. EXI(EX + EY), 0, w.p. EYI(EX + EY). * Eindhoven University of Technology, Department of Mathematics and Computing Science, P.O.Box 513, 5600 MB Eindhoven, The Netherlands. 1 Hence, denoting the Laplace--Stieltjes transforms of X, Z and T, resp., by X"'('), Z"'(·) and T"'(·), we have, because Z is now the waiting time in the MIGI1 queue,