A queueing system subject to breakdown and having non-stationary Poisson arrivals
Stanford University. Applied Mathematics and Statistics Laboratory, Andrew W. Shogan, United States. Office of Naval Research, Stanford University. Dept. of Operations Research, Stanford University. Dept. of Statistics
Dept. of Operations Research and Dept. of Statistics, Stanford University, 1977 - Queuing theory - 23 pages
This paper considers a single server queueing system that alternates stochastically between two states: operational and failed. When operational, the system functions as an M/Ek/1 queue. When the system is failed, no service takes place but customers continue to arrive according to a Poisson process; however, the arrival rate is different from that when the system is operational. Thus, both the arrival and service distributions are nonstationary. The durations of the operating and failed periods are exponential with mean 1/c-alpha and Erlang with mean 1/c-beta, respectively. Generating functions are used to derive the steady-state quantities L and W, both of which are decreasing and convex functions of c. The paper includes an analysis of several special and extreme cases and an application to a production-storage system. (Author).
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arrive according average number Avi-Itzhak and Naor backlogged controls how rapidly convex functions corresponding quantities customer arrived decreasing and convex denotes the number DERIVATION OF H(y Erlang distribution Erlang with mean expected number expected value exponentially distributed failed periods GQ(z k+1 m+1 l-yk M/E l queue M/E^l queue mean l/cp NON-STATIONARY POISSON NONSTATIONARY POISSON ARRIVALS November 7 I977 number of customers number of phases number of service operational and failed OPERATIONS RESEARCH PAOE Poisson arrival process Poisson process production-storage model production-storage system queue first becomes queue not subject queue with parameters queueing model QUEUEING SYSTEM SUBJECT random variable recursively to express repair period repair phases remaining REPORT NUMBER Section SECURITY CLASSIFICATION service and repair service phases service rate shape parameter Shogan special and extreme stationary steady state quantities steady-state probabilities subject to breakdown system is failed system is operational system occilates Yechiali and Naor yields Z._n