Accession Number : AD0720325

Title :   Extreme Values in the GI/G/1 Queue.

Descriptive Note : Technical rept.,

Corporate Author : STANFORD UNIV CALIF DEPT OF OPERATIONS RESEARCH

Personal Author(s) : Iglehart,Donald L.

Report Date : FEB 1971

Pagination or Media Count : 19

Abstract : Consider a GI/G/1 queue in which Wn is the waiting time of the nth customer, W(t) is the virtual waiting time at time t, and Q(t) is the number of customers in the system at time t. Let the extreme values of these processes be Wn* = max(Wj: 1= or < j = or <n), W*(t) = sup (W(s): 0 = or < s = or < t), and Q*(t) = sup (Q(s): 0 = or < s = or < t). The asymptotic behavior of the queue is determined by the traffic intensity rho, the ratio of arrival rate to service rate. When rho<1 and the service time has an exponential tail, limit theorems are obtained for Wn* and W*(t); they grow like log n or log t. When rho = or > 1, limit theorems are obtained for Wn*, W*(t), and Q*(t); they grow like n to the 1/2 power or t to the 1/2 power if rho = 1 and like n or t when rho > 1. For the case rho < 1, it is necessary to obtain the tail behavior of the maximum of a random walk with negative drift before it first enters the set from, but not including minus infinity, to zero. (Author)

Descriptors :   (*QUEUEING THEORY, DISTRIBUTION FUNCTIONS), STOCHASTIC PROCESSES, EXPONENTIAL FUNCTIONS, THEOREMS

Subject Categories : Operations Research

Distribution Statement : APPROVED FOR PUBLIC RELEASE