A stochastic process whose state space is the set {0,1,2,3,...} where the value corresponds to the number of customers i

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A Stochastic Process Whose State Space Is The Set 0 1 2 3 Where The Value Corresponds To The Number Of Customers I 1 (150.35 KiB) Viewed 26 times

A stochastic process whose state space is the set {0,1,2,3,...} where the value corresponds to the number of customers in the system, including any currently in service. Arrivals occur at rate , according to a Poisson process and move the process from state i to i + 1. Service times have an exponential distribution with rate parameter u in the M/M/1 queue, where 1/u is the mean service time. All arrival times and services times are (usually) assumed to be independent of one another. A single server serves customers one at a time from the front of the queue, according to a first-come, first served discipline. When the service is complete the customer leaves the queue and the number of customers in the system reduces by one. The buffer is of infinite size, so there is no limit on the number of customers it can contain. a. What is the limiting distribution for large values of time t? [2] b. The expected number of customers in the system in the long run? [2] C. The expected proportion of time that there are no customers in the system?[1] d. The expected proportion of time that there are customers in the system? [1] e. Expected total time that a customer spends in the system, assuming that the process has been running for a long time [5]