Cinema X has two entrances on opposite sides of the streets (so queues are independent). There is one server who sells ticket at each of these entrances. The arrival rate to each of the queues is the same and is equal to 8 customers per hour. One server is faster than the other one. Service time for server A is 4 minutes on average for each customer while server B can serve on average 20 customers per hour. Assume all the inter-arrival times and service times are exponentially distributed.
a). On average, how many customers are waiting (not including those receiving service) in cinema X?
b). The manager of Cinema X is considering closing down the slower server. Assume that in this case the faster server would serve all 16 arrivals per hour and the service rate does not change (the inter-arrival times and service times are still exponentially distributed). To choose between these options, the manager needs to do the cost analysis. Assume that the manager has estimated that waiting in line is costly in terms of dis-utility of customers and each customer incurs $10 cost in an hour wait in the line. On the other hand, the manager pays $7 to each server. Given this information can you help the manager to make its decision