Question

(1) Consider a "two machine two repairman" problem, where each machine independently breaks down at rate μ (that is after an exponential waiting time with parameter μ) (a) If each repairman repairs a machine at rate λ, calculate the long term proportion of time when both machines are broken. [Hint: calculate the stationary probabilities.] You can also consider 4 marks] (b) Assume that repairman Abel works at rate X, and repairman Bernard works at rate Xb. When there is only one broken machine the repairmen flip a fair coin to decide which of them will fix it. Calculate again the long term proportion of time when both machines are broken. (Hint: use state space S 10, la, 1b, 2), where la represents one broken machine with Abel working [15 marks] this problem as a special case of the second part of the question. on it, and 1b represents one broken machine with Bernard working on it.)

Answer 1

A)

his is not a birth and death process since we need more information than just the number working. We also must know which machine is working. We can analyze it by letting the states be
B: both machines are working
1: 1 is working, 2 is down and being repaired
2: 2 is working, 1 is down and being repaired
01: both are down, 1 is being repaired
02: both are down, 2 is being repaired

VB=
2

V1=$\large \mu 1+ \mu$

V2=$\large \mu 2+ \mu$

P_b,1=
/(1 2)

P_P1,b= $\large \mu /( \mu 1+ \mu )$

P_2,b=

p/(p2 + 11)

P_01,1=p_02,2=1