1. Chapter 2 Machine Interference Model
Long Run Analysis
Deterministic Model
Markov Model

2. Problem Description Group of m automatic machines
Operator must change tools or perform minor repairs
How many machines should be assigned to one operator?
Performance measures
Operator utilization:  = fraction of time the operator is busy
Production rate: TH = # finished items per unit time
Machine availability:  = TH/G, where G is the gross production rate, or the production rate that would be achieved if each machine were always available
Note: In this queuing system, the machines are the customers!
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Chapter 2

3. Long Run Analysis Each machine has gross production rate h
Pn is the proportion of time that exactly n machines are down:
Then, given Pn,
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Chapter 2

4. Eliminate some unknowns
Suppose the mean time to repair a machine is 1/, and the mean time between failures for a single machine is 1/.
= avg. # of repairs in (0,t] = t
= avg. # of failures in (0,t] =
In the long run, assuming the system is stable,
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5. Queuing Measures of Performance
= average # of machines waiting for service
= average number of machines down
= average downtime duration of a machine
= average duration of waiting time for repair
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Chapter 2

6. Little’s Formula Observe from the previous equations: where
is the total number of failures per unit time
= the arrival rate of customers to the queuing system
Little’s formula relates mean # of customers in system to mean time a customer spends in the system.
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7. A Deterministic Model
Suppose each machine spends exactly 1/ time units working followed by exactly 1/ time units in repair. Then if
and we could stagger the failure times, we would have no more than one machine unavailable at any time, so that
(Otherwise,
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8. A Markov Model
Let be the time between the (n-1)st and the nth failure of machine j, and
be the time duration of the nth repair
The time until the first failure is
N(t) = # of machines down at time t follows a CTMC with
S = {0, 1, …, m} and
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9. Steady-State Probabilities
satisfy the balance equations or level-crossing equations
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10. Solution
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