Optimal Scheduling Among Intermittently Unavailable Servers

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Optimal Scheduling Among Intermittently
Unavailable Servers

• Simon Martin Isi Mitrani
• University of Newcastle upon Tyne

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Model
m1, x1, h1
Arrivals l
policy
m2, x2, h2
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Parameters

• Arrival rate l
• At queue i (i1,2)
• Average service time 1/mi
• Average operative period 1/xi
• Average repair time 1/hi
• Average holding cost ci

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Problem

• A scheduling policy specifies, for every possible
  system state, whether an incoming job which finds
  that state is sent to queue 1 or to queue 2.

Find a policy that minimizes average holding
costs.
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Solution

• The problem is tackled using the tools of Markov
  decision theory.
• The optimal policy can be computed numerically by
  
• uniformizing the continuous time Markov process,
• replacing it with an equivalent discrete time
  Markov chain and
• truncating the state space to make it finite.

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Uniformization

• The instantaneous transition rates are modified,
  so that the transition rate out of any state is
  1, with the addition of transitions which do not
  change the current state.

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State of discrete time Markov chain

• S (i, j, b1, b2, a)
• Number of jobs in server 1 i
• Number of jobs in server 2 j
• Availability of server 1 b1
• Availability of server 2 b2
• Arrival event a

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Stationary policy for minimizing total average
discounted costs over an infinite horizon
This equation can be solved iteratively. The
interesting case is a ?8
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Minimize total average cost over a finite horizon
of n steps
Solve recurrences in n steps, starting from
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Steady-state average cost per step, independent
of the starting state
Obtained by simulation
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Policies examined

• Heuristic (smallest expected conditional holding
  cost per job)
• Random
• Selective (send only to operative servers
  N.Thomas)
• Shortest Queue
• Optimal (minimal steady-state cost per step)

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Varying l
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Varying m
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Varying x
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Varying Holding Cost