During summer months an air conditioner repair service adopts a policy
of accepting new
jobs on at most two air conditioners while working on one. Suppose a
repairperson had n
(n > 1) more jobs waiting after starting to work on an air
conditioner. Then the number
of waiting jobs after the start of the next job would be
(1) n-1 if no new jobs arrive during work on the current job,
(2) n if one new job arrives in the meantime, and
(3) n+1 if requests for two or more arrive.
Upon finishing work on any job, if no air conditioner is waiting for
service, this worker
has also adopted a policy of going off to do some other work for a
length of time that has
the same distribution as the service time of an air conditioner.
Therefore, for the purpose
of modeling, the case n = 0 can be treated exactly as n = 1. Further,
from experience, this
worker has found that during a service period the probability
distribution of repair job
arrivals is as follows:
Number of arrivals 0 1 ?2
Probability 0.3 0.5 0.2
Let Xn be the number of air conditioners waiting for service at the
end of the nth job. To
be consistent with definitions, define Xn to be the number actually
waiting soon after the
repairperson takes on a new job (which is excluded from Xn). Modeling {Xn} as a
Markov chain with discrete state space {0, 1, 2, ?0}, the transition
probability matrix
can be obtained as follows:
P00 = P(0 or 1 new job arrivals) = 0.3 +0.5 = 0.8
P01 = P(2 new job arrivals) = 0.2
Pi,i-1 = 0.3, Pii = 0.5, Pi,i+1 = 0.2, i > 0
Solve this problem in its general case by considering the job arrival
distribution given
below:
Number of arrivals 0 1 ?2
Probability p q r |
