Problem 4 (20 points): Tami stand and customer problems (slightly modied from HW#3) The arrival process of customers at a taxi stand is a Poisson process with rate A, and the arrival process of taxis at the stand is Poisson process with rate ,u.. Assume A 79 p. Let p = A/ a 1. An arriving customer who nds taxis waiting (being delayed) leave immediately in a taxi. An arriving taxi who nds customers waiting leave immediately with one customer each. Taxis queue up to a limit of L taxis, i.e., arriving taxis who find L other taxis waiting will leave immediately without a customer. Similarly, customers queue up to a limit of C, i.e., an arriving customer who finds C other customers waiting will leave immediately without a taxi. See Figure 1 below for illustration. size C Figure 1: System for Problem 4 (a) (10 points) Define an appropriate \"state\" for this problem to be Markov, and draw the state transition diagram with corresponding transition rates. Find the steady-state probability. (b) (4 points) What is the probability that an arriving taxi leaves immediately without a customer since there are L other taxis waiting? Your answers must be in terms of L, C and p only. (c) (5 points) Find the average number of customers waiting in the queue in terms of L, C and p only. No need to nd the closed form of any summation involved here