In a bank drive-through, there is a single service window and room only for two cars to line-up to wait for service. The mean time between arrivals for drive through customers is 5 minutes. The mean time to complete a customer transaction is 3 minutes. The number of arrivals is distributed according to a Poisson distribution and the service times are exponentially distributed. What is the probability that there are no vehicles in the system? On average how many cars are in the system? What is the probability that the system is full and new arrival must drive on? What is the average time a customer spends in waiting in line to be served?
Added by F-Tima M.
Step 1
This is an M/M/1/K queue with capacity K = 3 (1 in service + 2 waiting). λ = 1/5 = 0.2 customers/min, μ = 1/3 ≈ 0.333333 /min, ρ = λ/μ = 0.6. Show more…
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Assume that at a bank teller window, the customers arrive in their cars at an average rate of twenty per hour according to a Poisson distribution. Assume also that the bank teller spends an average of two minutes per customer to complete a service, and the service time is exponentially distributed. Customers, who arrive from an infinite population, are served on a first-come-first-served basis, and there is no limit to the possible queue length. a. What is the expected waiting time in the system per customer? b. What is the mean number of customers waiting in the system? c. What is the probability of zero customers in the system? d. What value is the traffic intensity?
Customers arrive at a bank counter manned by a single cashier according to a Poisson distribution with a mean arrival rate of customers per hour. The cashier attends to the customers on a first-come, first-serve basis at an average rate of customers per hour, with the service time exponentially distributed. Find the probability of the number of arrivals (0 through 5) during (i) a 5-minute interval and (ii) a 30-minute interval. Also, find the probability that the queuing system is idle, the probability associated with the number of customers (0 through 5) in the queuing system, the probability that there are more than customers in the queuing system, the time a customer should spend in the queue, and the time a customer spends before leaving the bank counter.
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