Chapter 20.9, Problem 6P

Explanation of Solution

To Prove:

For a machine repair model, prove that

$W=\frac{K}{\overline{\lambda }}-\frac{1}{\lambda }$

Proof:

A machine repair problem containing “K” machines and “R” repair people is a $M\left|M\right|R\left|GD\right|K|K$queuing system where the inter arrival times, service times are exponential, servers, general queue discipline, first “K” implies that at any time, no more than “K” machines may be present, the second “K” implies that the arrivals are drawn from a finite source of size “K”.

The steady state probability at state j is as follows:

$\begin{array}{l}{\pi }_j=(C{}_j){\rho }^j{\pi }_0\text{ (j=0,1,}\mathrm{....}\text{,R)}\\ \text{ = }\frac{(C{}_j){\rho }^{j}j!{\pi }_0}{R!R^{(j-R)}}\text{ (j=R+1,R+2,}\mathrm{...}\end{array}$