Mathematics — Probability · ergodic-markov-chains-queueing
Problem
A discrete time queueing system of capacity n consists of the person being served and those waiting to be served. The queue length x is observed each second. If 0<x<n, then with probability p, the queue size is increased by one by an arrival and, independently, with probability r, it is decreased by one because the person being served finishes service. If x=0, only an arrival (with probability p) is possible. If x=n, an arrival will depart without waiting for service, and so only the departure (with probability r) of the person being served is possible. Form a Markov chain with states given by the number of customers in the queue. Modify the program FixedVector so that you can input n, p, and r, and the program will construct the transition matrix and compute the fixed vector. The quantity s=p/r is called the traffic intensity. Describe the differences in the fixed vectors according as s<1, s=1, or s>1.