Operations Management - Please help with all parts (Its only 1 question)

1. Consider a mall retail shop that has a single server, who can service customers at a rate of 20 per hour. Customers nominally arrive at the retail shop at a rate of 40 per hour. However, these customers are impatient shoppers and have an exponential balk rate as the line gets longer. Specifically, the arrival rate is given by the following equation: R=0,1.... to = 40x ( (a) Draw a rate diagram of the retail shop's queuing system. In your rate diagram, include at least n = 0,1,2,3,4,5,6. (b) Calculate the probability po as a function on po. Although the retail shop queue is an infinite queue, the equations get pretty messy. However, in your answer to part (b), you should observe that po, and for the matter any pn in which n = 6,7..., is going to be extremely small. Consequently, for the remainder of this problem, we will assume that pr0 for n = 6,7...., so we can approximate the queue as finite. c) Based upon the finite queue assumption, calculate the probability distribution of the number of customers in the retail shop queuing system for n = 0...,5. (d) Calculate the expected number of customers in the retail shop queuing system. (e) Calculate the expected arrival rate of customers to the retail shop queue. (f) Calculate the expected rate of balking customers per hour. () Calculate the expected throughput time in the retail shop queuing system. (h) Calculate the expected waiting time in the retail shop queue, not including the time being served. (i) Calculate the expected number of customers waiting in the retail shop queue, not including the customers being served. 1. Consider a mall retail shop that has a single server, who can service customers at a rate of 20 per hour. Customers nominally arrive at the retail shop at a rate of 40 per hour. However, these customers are impatient shoppers and have an exponential balk rate as the line gets longer. Specifically, the arrival rate is given by the following equation: R=0,1.... to = 40x ( (a) Draw a rate diagram of the retail shop's queuing system. In your rate diagram, include at least n = 0,1,2,3,4,5,6. (b) Calculate the probability po as a function on po. Although the retail shop queue is an infinite queue, the equations get pretty messy. However, in your answer to part (b), you should observe that po, and for the matter any pn in which n = 6,7..., is going to be extremely small. Consequently, for the remainder of this problem, we will assume that pr0 for n = 6,7...., so we can approximate the queue as finite. c) Based upon the finite queue assumption, calculate the probability distribution of the number of customers in the retail shop queuing system for n = 0...,5. (d) Calculate the expected number of customers in the retail shop queuing system. (e) Calculate the expected arrival rate of customers to the retail shop queue. (f) Calculate the expected rate of balking customers per hour. () Calculate the expected throughput time in the retail shop queuing system. (h) Calculate the expected waiting time in the retail shop queue, not including the time being served. (i) Calculate the expected number of customers waiting in the retail shop queue, not including the customers being served