Problem 1. Consider a fast food restaurant where customers arrive and get in line according to...
Problem 1. Consider a fast food restaurant where customers arrive and get in line according to a Poisson process with an average rate of 120 customers per hour. The restaurant has one line and the amount of time it takes the cashier to serve a customer is exponentially distributed with a mean of 2 minutes. Let X_t denote the number of customers in line at time t.
1. Give the state space of the chain (X_t)t≥0.
2. For each state i,
a) find the hold time parameter i,
(b) find the embedded chain jump probabilities P_ij , (c) find the transition rates q_ij .
3. Make a transition rate diagram for the chain.
4. Assume that there are initially no customers in the system. Find the probability that there are 2 customers after 5.5 minutes
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Problem 1. Consider a fast food restaurant where customers
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