Verify the following relationships for an M/M/1 queueing system: 17.6-5.

It is necessary to determine how much in-process storage space to allocate to a particular work center in a new factory.

Jobs arrive at this work center according to a Poisson process with a mean rate of 3 per hour, and the time required to perform the necessary work has an exponential distribution with a mean of 0.25 hour. Whenever the waiting jobs require more in-process storage space than has been allocated, the excess jobs are stored temporarily in a less convenient location. If each job requires 1 square foot of floor space while it is in in-process storage at the work center, how much space must be provided to accommodate all waiting jobs

(a) 50 percent of the time,

(b) 90 percent of the time, and

(c) 99 percent of the time? Derive an analytical expression to answer these three questions. Hint: The sum of a geometric series is N

n0 xn 1

1





xN x

1

.