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 94 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

Answer 1

Answers are in BOLD

	Serving time=		94.00	secs
	Lamda = arrival rate (poisson) = 1 in 2 mins= 1/2	0.50	per min
	Mu = mean service rate (exponential) = 60/serving time	0.64	per min
	Serving time= 1/Mu=	1.567	mins
	Utilization = rho = lamda / mu =	0.7833
Answer c	Average no. in queue:
	Average no queue =Lq = (rho)2 / (1-rho)
	Average no of customers waiting in line =Lq =	2.83	customers
	Average time spent in queue:
	Wq = waiting time in the line for service = Lq / lambda =	5.6641	mins
Answer a	Average time spent in the system:
	W = average time in the system = Wq + serving time =	7.23~7	mins
Answer b	Probability of zero customers in system:
	P (0)= 1-rho		0.22

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