In a departmental store one cashier is there to serve the customers. And the customers pick up their needs by themselves. The arrival rate is 9 customers for every 5 minutes and the cashier can serve 10 customers in 5 minutes. Assuming Poisson arrival rat

(a) Average number of customers in the system can be calculated using Little's Law: Average number of customers in the system = arrival rate * average time in the system To find the average time in the system, we need to find the sum of the average time a customer spends in the queue and the average time a customer spends being served. Average time in the system = average time in queue + average service time We can use the formula for average service time for an exponential distribution: Average service time = 1 / service rate = 1 / (10 customers/5 minutes) = 0.5 minutes To find the average time in the queue, we can use the formula: Average time in queue = (arrival rate * service rate) / (service rate - arrival rate) * service time Plugging in the values we have: Average time in queue = (9 customers/5 minutes * 10 customers/5 minutes) / (10 customers/5 minutes - 9 customers/5 minutes) * 0.5 minutes = 0.45 minutes So the average time in the system is 0.5 + 0.45 = 0.95 minutes. Plugging this into Little's Law: Average number of customers in the system = 9 customers/5 minutes * 0.95 minutes = 17.1 So the average number of customers in the system is 17.1. (b) Average number of customers in the queue can be found using the formula: Average number of customers in the queue = (arrival rate)2 / (service rate * (service rate - arrival rate)) Plugging in the values we have: Average number of customers in the queue = (9 customers/5 minutes)2 / (10 customers/5 minutes * (10 customers/5 minutes - 9 customers/5 minutes)) = 0.405 So the average queue length is 0.405 customers. (c) Average time a customer spends in the system is already calculated in part (a) and is 0.95 minutes. (d) Average time a customer waits before being served can be found by subtracting average service time from average time in queue: Average time a customer waits = average time in queue - average service time = 0.45 minutes So the average time a customer waits before being served is 0.45 minutes.