Chapter 2 Machine Interference Model

1. Chapter 2Machine 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! 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, 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, Chapter 2

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 Chapter 2

6. Little’s Formula Observe from the previous equations: where is the total average 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. Chapter 2

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, Chapter 2

8. A Markov Model Let be the time between the (n-1)st repair and the nth failure of machine j, and be the time duration of the nth repair (indep.) The time until the first failure is N(t) = # of machines down at time t follows a CTMC with S = {0, 1, …, m} and Chapter 2

9. Steady-State Probabilities satisfy the balance equations or level-crossing equations Chapter 2

10. Solution Chapter 2

11. Erlang Distribution If failure and/or repair times are not exponential, can fit an Erlang distribution by matching moments: Big advantage: Can still model as a CTMC. Consider time to machine failure (each machine) as Erlangk. Can think in terms of k phases in the time to failure, where the time the m/c spends in each phase is exponential (kl): Mean time spent in each phase = Mean total time to failure = Chapter 2

12. 2l 2l 0;1 1 0;2 m Expanded State Definition Mi(t) = # of machines operating in phase i at time t For example, if k = 2, then a single machine without interference follows the CTMC (1 = failed state): Chapter 2

13. Transitions among States (k=2) Steady state probabilities: Rate into state 2l(l1+1) m 2l(l2+1) Chapter 2

14. Balance Equations This system of equations (for any k) has the solution: Chapter 2

15. SS Number of Machines Working From the previous equation and get Find probabilities by normalizing to 1. This distribution is independent of k or any other characteristics of the failure time distribution. It can be shown that the same state distribution holds for any failure time distribution! Chapter 2