Question

The manager of post office must determine how many workers should work on Tuesdays. For every minute a customer stands in line, the manager believes that a delay cost of 7¢ is incurred. An average of 5 customers per minute arrive at the post office. On the average, it takes a worker 1 minute to complete service. It cost the post office $15 per hour to hire a worker. Interarrival times and service times are exponential. To minimize the sum of service costs and delay costs, how many workers should the post office have working on Tuesdays?

Answer 1

Arrival rate = lambda = 5 per min

Service rate mu = 1 per min.

Utilisation (rho) =5

Since the utilisation is 5 , there must be more than 5 workers to serve.

For analysis we will start with 6 workers and keep on increasing the number of workers .to find optimum cost.

Case I - 6 workers

Number of customers wating with rho =5 and 6 workers

Lq = 2.9376

Waiting time in line Wq = Lq /lambda = 2.9376 /5 =0.5875 minutes

Total waiting time per minute for all customers = 0.5875 xlambda = 0.5875x5 = 2.9376 minutes

cost of waiting per minute for all cuatomers . =0.07 x2.9376 =$0.2056

Hourly cost of waiting = 0.2056x60 =12.338

Total cost ( waiting cost + wages) = 15x6+12.3379 =102.338

Case II - number of customers waiting with 7 workers  and rho =5

Lq =0.8104

Waiting time for one customer= 0.8104 /5 =0.16208 min.

Waiting time per minute for all customers = 0.16208x5 =0.8104 min.

cost of waiting per min. = 0.07 dollars  x0.8104 =$0.05673

Hourly cost of waiting = 0.05673 x60 =3.4036

Total cost ( waiting cost + wages of 7 workers  )= 15x7+3.4036 = 108.4036

There is no point in calculating  further with higher number of workers as the waiting cost is never more than $15, the cost of adding one more resource.

6 workers option has lowest cost of 102.338, hence the company should go with 6 workers.