On the expected number of crossings of a level in certain stochastic processes

We consider a stochastic process which increases and decreases by simple jumps as well as smoothly. The rate of smooth increase and decrease with time is a function of the state of the process. The process is not constant in time except when in the zero state. For such processes a relation is derived between the expected number of true crossings (as oppcsed to skippings by which we mean vertical crossings due to jumps) of a level x, say, and the time dependent distribution of the process. This result is applied to the virtual waiting time process of the GI/G/1 queue, where it is of particular interest when the zero level is considered, as the underlying crossing process is then a renewal process. It leads to a new derivation of the busy period distribution for this system. This serves as an example for the last brief section, where an indication is given as to how this method may be applied to the GI/G/s queue. Naturally, the present method is most powerful when the original process is a Markov process, so that renewal processes are imbedded at all levels. For an application to the M/G/1 queue, see Roes [3]. 1. The expected number of level x crossings. Let {X(t), t ? 0} be a stochastic process on the non-negative real line [0, co). The process is supposed to be separable for closed sets; that is, its sample functions are left or right continuous. Further, the sample functions X(t, w), w eo are supposed to have a fairly simple structure on the subrange (0, o). Let T(wo) = {t: X(t, wo)> 0} and assume that the sample derivative exists for almost every t e T(wo). In addition, the absolute value of this derivative is r(x)> 0, whenever X(t, wo)= x> 0. A typical sample path is shown in Figure 1. In the sequel we will need to distinguish between crossings and skippings of a level x, say. The level x is skipped at t = t1 if X(t1 -) ~ x 5 X(t1 +). If in the latter equalities are satisfied the level x is ossed; this may be done from above as at t2 or below as at t3. It will be more convenient to deal with processes for which r(x) = 1 and we therefore transform the range through division by r(x). We now consider a typical sample path as shown in Figure 2 and define N(Tx,ow) as the number of times the path actually crosses the line X(t, w) = x > 0 in the Received 30 January 1970. 766 This content downloaded from 207.46.13.124 on Wed, 22 Jun 2016 05:25:13 UTC All use subject to http://about.jstor.org/terms On the expected number of crossings of a level in certain stochastic processes 767