A probability limit theorem with application to a generalisation of queueing theory

Single server queues with modified service mechanisms

“…As in [2] we may show that the limiting distributions Q n = ]im m _ oo Q n (m) and R n = lim m _ 0O R n {m) both exist and are equal. We shall now prove …”

“…Finch [2] has shown for his process that this is the stationarity condition, and it is also a necessary and sufficient condition for the existence of a stationary distribution for our process. The proof extends Lindley's [5] argument in a similar manner to that in [2], and will not be given here. The condition is independent of the service time distribution for a customer joining a non-empty queue; we may intuitively expect this, for regardless of the non-zero finite size of the service time for a customer finding the queue empty, the waiting time must eventually reduce to zero again with probability unity if E(u) < 0, and may build up indefinitely if E(u) ^ 0.…”

“…U>n + Un ^n + "" > 0, W n > 0 c B c n > 0, w n = 0 0 otherwise, where u n = s n -t n+l and c n = r n -t n+1 ; {u n } and {c n } are independently and identically distributed random variables with the d.f. 's (2) U(x) = J~B(* + y)dA{y), C{x) = j~D{x + y)dA(y), G. F. Yeo [2] with expectations…”

“…Finch [2] has considered a process which differs from ours only when the w-th arrival joins an empty queue; the customer then waits for a time v n before commencing a service of length s n , instead of receiving an immediate service of length r n . If we put D[x) = $%B(x -y)dV(y), where V(x) is the d.f.…”

AGI/M/1 queue with a modified service mechanism

“…For the standard GI]M]I system (~=t~) our results yield some interesting facts about the limiting queue length distribution. In Section 5 we show that the methods of Section 2 yield results for the GI/M/1 version of Finch's model [2]. …”