Question

What are the operating characteristics of a waiting
line system? Discuss.

A repairman is to be hired by a company to repair machines that
break down randomly with an average of rate of four per hour for an
8-hour day. The machines wait in the queue if necessary and are
repaired on a first- come- first served basis. The cost of
non-productive machine time is Rs 90 per hour. The company has the
option of choosing either a fast or a slow repairman. The fast
repairman charges Rs 70 per hour and will repair machines at an
average rate of 7 per hour, while the slow repairman charges Rs 50
per hour and will repair machines at an average rate of 6 per hour.
Assuming that the pattern of machine’s breakdown is Poisson
distributed and the repair time is exponentially distributed. Using
the above information compare the total expected daily cost for
both repairmen and advise management which repairman should be
hired?

Answer #1

The operating characteristics of a waiting line system:

- The arrival rate: the rate at which entities arrive at the system
- The service rate: the rate at which entities get served at the system
- The waiting rate: the amount of time, entities wait in a queue

Given,

lambda (l)= 4 units * 8 hours = 32 machines per day

Let the fast repairman be A. Let the slow workman be B.

_{mA}=56 machines per day

_{mB}=48 machines per day

Probability for the repairman to be idle is given by P =1–r

And r= l/m

So, P_{A}= 1- l/ m_{A} = 1- 32/56 =0.429

Expected idle time for A= 0.429*8 = 3.432 hour

Expected work time for A= 8-3.432=4.568 hour

Total cost per day of A = (90*3.432) + (70*4.568) = Rs 628.64

P_{B}= 1- l/ m_{B} = 1- 32/48 = 0.333

Expected idle time for B= 0.333*8 =2.664 hour

Expected work time for A= 8-2.664=5.336 hour

Total cost per day of B= (90*2.664) + (50*5.336) = Rs 506.56

As, both the fast and the slow repairman is able to handle the jobs at the company but the cost of slow repairman is comparatively cheap.

So, based on cost effectiveness, the slow repairman must be hired.

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