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 87 seconds completing his or her transactions. Transaction time is exponentially distributed. Determine the following: a. The average time customers spend at the machine, including waiting in line and completing transactions b. The probability that a customer will not have to wait upon arriving at the automatic teller machine c. The average number waiting to use the machine

The Peachtree Airport in Atlanta serves light aircraft.  It has a single runway and one air traffic controller to land planes.  It takes an airplane 12 minutes to land and clear the runway (following an exponential distribution).  Planes arrive at the airport at the rate of one every 15 minutes.  The arrival rate follows a Poisson distribution.  A plane is considered to have entered the “system” once it has notified the airport that it is in the vicinity and wants to land.   For purposes of this analysis, you can ignore the planes taking off.   a) Determine the average number of planes that are in the system waiting to land.         b) Find the average time a plane is in the system either waiting to land or landing.           c) What is the probability that a plane approaching the airport will find at least two other that are already in the “system” that is waiting to land?         d) Suppose that the cost assigned to a plane while it is waiting to land (in the stack) is $1,000…

Town Taxi uses three dispatchers to handle requests for service and to dispatch the cabs. The telephone calls that are made to Town Taxi use a common number. When all dispatchers are busy, the caller hears a waiting song until the first dispatchers becomes available. The telephone system will assign available dispatcher to the waiting callers according to FCFS. Assume that the arrival of calls follows a Poisson distribution, with an average of 35 calls per hour. Also assume that the time that each dispatcher spends with a customer follows Exponential distribution and each dispatcher takes an average of 3 minutes with each caller. 15.What is the average time in minutes that a customer spends in the system? 3.8 mins 7 mins 4.8 mins 8.9 mins

A firm is engaged in both shipping and receiving activities. The management is always interested in improving the efficiency of new innovations in loading and unloading procedures. The arrival distribution of trucks is found to be Poisson with arrival rate of 3 trucks per The service time distribution is exponential with unloading rate of 4 trucks per hour. Determine (i) expected number of trucks in the queue, hour. (ii) expected waiting time of the truck in the queue, (iii) probability that the loading and unloading dock and workers will be idle, (iv) what reductions in waiting time are possible if loading and unloading is standardised?