Many of a bank’s customers use its automatic teller machine to transact business

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 91 seconds completing his or her transactions. Transaction time is exponentially distributed.

a. Determine the average time customers spend at the machine, including waiting in line and completing transactions. (Do not round intermediate calculations. Round your answer to the nearest whole number.)
  
Average time             minutes

b. Determine the probability that a customer will not have to wait upon arriving at the automatic teller machine. (Round your answer to 2 decimal places.)
  
Probability             

c. Determine the average number of customers waiting to use the machine. (Round your answer to 2 decimal places.)
  
Average number             customers

Explanation / Answer

Solution-

This queue system is called M/M/1 system. Its a single server with poisson arrival rate and exponential service rate.

Given data,

R, arrival rate = 1 every 2 mins = 30 per hour

S, service rate = 1 every 91 seconds = (3600/91) per hour = 39.56 per hour

a. Ws or average waiting time = 1/(S - R) = 1/(39.56-30) = 0.1046 hour = 6.276 mins = 6 mins

b. Probability that a customer will not have to wait upon arriving at the automatic teller machine, P0 = 1 - (R/S)

= 1 - (30/39.56) =  0.24

c. the average number of customers waiting to use the machine = R^2/ (S(S - R)) = 30^2/39.56*(39.56-30) =  2.38 customers in line  

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