Regenerative Processes

Abstract

The GI |GI|1|∞ model.Consider the model discussed in Section 4.3.1 and denote by Q(t) the number of customers occupying the system at time t. The trajectory Q(t), t ≥ 0, depends on “governing” sequences e = (e ₀, e _l,...) and s = (s ₁, s ₂,...) of interarrival and service times, respectively. Denote this dependence by:

$$Q\left( \cdot \right) = f\left( {e,s} \right)$$

(1)

where f is a mapping of the set of pairs of sequences (e, s) to the set of functions Q(t), t ≥ O. It is tedious to write down f in a closed form. However, fortunately, this is not necessary. Suppose that T _k is the time at which the k-th busy period starts. Then Q(t) jumps up at t = T _k from 0 to 1, and T _k is the beginning of both current interarrival and service times. It follows that the “shifted process” Q(T _k + .) depends on the values e _N+1, e _N+2,...) and s _N+1, s _N+2, ..., where N is the number of customers having been served up to the time T _k similarly to the dependence of the process Q(•) on (e, s):

$$Q\left( {{T_k} + \cdot } \right) = f\left( {\left( {{e_{N + 1}},{e_{N + 2}}, \ldots ,} \right)\left( {{s_{N + 1}},{s_{N + 2}}, \ldots ,} \right)} \right).$$

(2)

Here, f is the same mapping as in (1). Besides, the r.v. N does not depend on both (e _N+1, e _N+2,...) and (s _N+1 s _N+2,...). Thus, Q(T _k + •) has the same distribution as Q(•), and does not depend on the “prehistory” Q(t),t ≤ T _k . It follows that the whole process Q(•) can be “cut” by the moments {T _k } on i.i.d. “fragments” which we shall call cycles. Namely, let us define a cycle as a pair comprising a random variable X ≥ 0 and a random proccess Q ^c(t), 0 ≤ t < X. The r.v. X can be viewed as the duration of a busy period plus the length of the following idle period, and Q ^c(•) as the corresponding queue-length process within these busy and idle periods. Let (Q ^c _i (•), X _i be a sequence of i.i.d. copies of (Q ^c(•), X). Then the process Q(•) can be presented in the form:

$$Q\left( t \right) = Q_i^c\left( {t - {T_{i - 1}}} \right), {T_{i - 1}} \leqslant t \leqslant {T_i}$$

(3)

where T ₀= 0, T _i =X ₁+ ••• + X _i , i ≥ 1.