×

Idle and busy periods in stable \(\text{M}/\text{M}/k\) queues. (English) Zbl 0930.60078

The first subject is the number \(N\) of arrivals in a stable \(\text{M}/\text{M}/k\) queue during an idle period, i.e. the time during which at least one of the \(k\) servers is idle. By deriving equations for the probability generating functions (p.g.f.) of the number \(N_j\), \(j=0,\dots, k-1\), of arrivals of an idle period that starts with \(j\) servers busy, explicit formulae for the p.g.f., the mean and variance are given. Then for the p.g.f. of the number of arrivals during a first passage time (from \(j\) to \(j+1\)) a \(j\)-fold continued fraction is given. In the last part, which is motivated by an overflow control problem, the results and similar ideas as well as renewal theory asymptotics are used to find the first and second moments of the portion of arrivals occuring while there are at least \(\xi\) customers waiting for a server. The asymptotic of the second moment for large \(\xi\) is established.
Reviewer: A.Brandt (Berlin)

MSC:

60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
PDFBibTeX XMLCite
Full Text: DOI