Chapter 2 Machine Interference Model

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Presentation transcript:

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

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! IE 512 Chapter 2

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, IE 512 Chapter 2

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, IE 512 Chapter 2

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 IE 512 Chapter 2

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. IE 512 Chapter 2

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, IE 512 Chapter 2

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 IE 512 Chapter 2

Steady-State Probabilities satisfy the balance equations or level-crossing equations IE 512 Chapter 2

Solution IE 512 Chapter 2