How large delays build up in a GI/G/1 queue
Let \(W_ k\) denote the waiting time of customer k, \(k\geq 0\), in an initially empty GI/G/1 queue. Customer O arrives at time 0 and customer k at time \(A_ 1+...+A_ k\). Let \(B_ i\), \(i=1,2,..\). be the service time \((A_ i\) and \(B_ i\) are i.i.d.), \(X_ k=B_ k-A_ k\), then \(X_ i\), \(i=1,2,..\). are i.i.d., and \(E\;X_ 1=-\mu\), \(\mu >0\). There are considered queues for which Cramér type conditions hold for X, and queues for which X has a regularly varying positive tail. Let C denote C[0,1], the space of continuous functions on the unit interval, and let D denote D[0,1], the space of right continuous functions with left limits on the unit interval, both with the topology of uniform convergence. Let \(w_ n(\cdot)\in C\) denote the polygonal path, and \(\hat w_ n(\cdot)\in D\) denote the piecewise continuous path, constructed from the points \((k/n,W_ k/n)\), \(0\leq k\leq n\). For a fixed \(a>O\) let \(P_ n\) denote the conditional distribution \((w_ n(\cdot)| W_ n>na)\) and \(\hat P_ n\) the conditional distribution \((\hat w_ n(\cdot)| W_ n>na)\). There are identified the weak limits of \(\{P_ n\}\) and \(\{\hat P_ n\}\) under certain moment conditions.
scientific article; zbMATH DE number 4138912