Question

Many of a bank’s customers use its automatic teller machine to transact business after normal banking hours. During the early evening hours in the summer months, customers arrive at a certain location at the rate of one every other minute. This can be modeled using a Poisson distribution. Each customer spends an average of 85 seconds completing his or her transactions. Transaction time is exponentially distributed.

Answer 1

Customers arrive at a rate = one every other minute (Poisson)

Mean service time = 90 seconds/customer (exponentially distributed)

M = 1

[Single Server, Exponential Service Time, M/M/1]

We must determine ? and ? first (we will use hours):

? = 1 customer/2 minutes x 60 minutes/hour = 30 customers/hour

? = 3600 minutes/hour / 90 seconds/customer = 40 customers/hour

a. Average time customers spend at the machine, wait time + transaction time (hours) (Ws):

W_s=W_q+1/?=0.075+1/40=0.075+0.025=0.100

b. Probability that a customer will not have to wait (P0):

c. Average number of customers waiting to use the machine (Lq):

Lq = 2.250 customers (see Part a above)

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