Problem on Poisson Distribution

1. Customers arrive at a two‐chair shoeshine stand at the rate of 10 per hour. The average duration of a shoeshine is 6 minutes. There is only one attendant, so that one chair is used as a waiting position. Customers who find both chairs occupied go away.

a. assuming Poisson input and exponential service times, compute the system state probabilities.

b. find the mean number of customers served per hour.

c. repeat the analysis to calculate the mean number of customers served per hour when there are two attendants at the stand (and no

waiting positions).

2. Speedy copy center located on Capitol Hill in Washington D.C. has three coin operated copying machines used primarily by U.S. Senators to make copies of their confidential diaries and illegal agreements with foreign governments and corporations. The owner, Hugh Makeham, is a former Xerox repairman and can fix a broken machine within 20 minutes on the average. Many times the machines are only jammed and he can get them working again quickly so the exponential will describe his repair time distribution. The time‐to‐breakdown of a freshly repaired copier averages 60 minutes and is also exponentially distributed.His revenue when all machines are working averages $80/hr per machine. When one machine is broken his revenue drops to $60/hr per working machine because of balking. When two machines are broken his revenue from the remaining machine decreases to $50/hr. When all machines are broken his revenue naturally drops to zero.

a) Compute his average hourly revenue.

b) If he employs a second, equally competent, repairman, what will be his total expected hourly income?

c) What is the largest hourly wage he can pay the employee without decreasing the expected hourly income he earned when working alone.

3. The sketch below shows an open queuing system with three stations. External arrivals to stations one to three occur at the Poisson rates of 5/hr, 7/hr and 10/hr respectively. 70% of the arrivals to workstation 1 leave the system while 30% proceed to work station 2. 10% of the workstation 2 arrivals are recycled back to workstation 1 while 90% proceed to work station 3. 85% complete service at workstation 3 and leave the system while 15% cycle back to workstation 2. The characteristics and service rates for each queue are given in the table below. Compute the number of customers waiting at each work station and the overall average waiting time in the system. Station Queue Type Service rate per server

1 M/M/1 10/hr

2 M/M/2 7/hr

3 M/M/3 9/hr