(Barke) Consider an M/M/1, M/M/e or M/M/ quese with arrival rate. In the steady state, the...

  

 

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(Barke) Consider an M/M/1, M/M/e or M/M/ quese with arrival rate. In the steady state, the deportarea from the system forma Poisson process with rate X, independently of a (so long as X). Furthermore, at time t the number of customers in the quese is independent of the departure process prior to time t. Problem 4 Consider a two-stage tandem network composed of two nodes with service rates and respectively. This means that a customer joins the end of the walt line at the first counter upon arrival and moves to the end of the wait line for the second counter after being served at the first. The external arrival rate is A (A<=1,2) and the arrival process is Poison. Assume that the service times at each node are exponentially distributed and mutually independent, and independent of the arrival processes, the total number of visitors is not bounded. 0,0 1,0 2,0 3,0 0.1 1.1 2.1 0,2 1,2 (a) 5 pts Give an example of a real life occurence of such a queuring system. 0,3 and (b) [15 pts] Show that the steady-state probabilities are given by Pnm = (1-P₁)pf (1 – P2)p, where p₁ = A (c)* [15 pts] Give a generalization of (b) to the case of n counters. P2 || (Barke) Consider an M/M/1, M/M/e or M/M/ quese with arrival rate. In the steady state, the deportarea from the system forma Poisson process with rate X, independently of a (so long as X). Furthermore, at time t the number of customers in the quese is independent of the departure process prior to time t. Problem 4 Consider a two-stage tandem network composed of two nodes with service rates and respectively. This means that a customer joins the end of the walt line at the first counter upon arrival and moves to the end of the wait line for the second counter after being served at the first. The external arrival rate is A (A<=1,2) and the arrival process is Poison. Assume that the service times at each node are exponentially distributed and mutually independent, and independent of the arrival processes, the total number of visitors is not bounded. 0,0 1,0 2,0 3,0 0.1 1.1 2.1 0,2 1,2 (a) 5 pts Give an example of a real life occurence of such a queuring system. 0,3 and (b) [15 pts] Show that the steady-state probabilities are given by Pnm = (1-P₁)pf (1 – P2)p, where p₁ = A (c)* [15 pts] Give a generalization of (b) to the case of n counters. P2 ||