## Approximate Stochastic Behavior of n-Server Service Systems with Large nFor many stochastic service systems, service capacities large enough to serve some given customer demand is achieved simply by providing multiple servers of low capacity; for example, toll plazas have many toll collectors, banks have many t- lers, bus lines have many buses, etc. If queueing exists and the typical queue size is large compared with the number n of servers, all servers are kept busy most of the time and the service behaves like some "effective" single server wit:l mean se.- vice time lin times that of an actual server. The behavior of the queueing system can be described, at least approximately, by use of known results from the much studied single-channel queueing system. For n» 1 , however, (we are thinking p- ticularlyof cases in which n ~ 10), the system may be rather congested and quite sensitive to variations in demand even when the average queue is small compared with n. The behavior of such a system will, generally, differ quite significantly from any "equivalent" single-server system. The following study deals with what, in the customary classification of queueing systems, is called the G/G/n system; n servers in parallel with independent s- vice times serving a fairly general type of customer arrival process. rhe arrival rate of customers may be time-dependent; particular attention is given to time - pendence typical of a "rush hour" in which the arrival rate has a single maximum possibly exceeding the capacity of the service. |

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### Contents

General Formulation | 1 |

Approximation Methods | 24 |

Approximations for Short Service Times | 57 |

Copyright | |

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### Common terms and phrases

Ac(t accurate analysis apply approximately normal arrival process arrival rate available servers batches become behavior Chapter compared with E{s conditional distribution constant curve E{A customer arrivals decreases described deterministic approximations distribution of N(t duration enter service equation equilibrium distribution estimate evaluated expected number exponential distribution fluctuations formulas geometric distribution graph graphical idle servers increase large compared least M/M/n system Markov process mean service methods N(Tq nearly negative normal approximation normal distribution number of arrivals number of customers number of servers order E{s parameter Poisson distribution Poisson process probability density problem properties qualitative queue of customers queueing starts queueing theory random service range renewal theory rush hour scale second term second transition Section Seiten service completions service distribution service system slowly varying small compared smooth standard deviation statistically independent stochastic correction stochastic effects stochastic queueing tion traffic intensity types typical values of N(t Var{N(t variance