Optimal Scheduling Among Intermittently Unavailable Servers

1. Optimal Scheduling Among Intermittently Unavailable Servers Simon Martin & Isi Mitrani University of Newcastle upon Tyne

2. Model m1, x1, h1 Arrivals l policy m2, x2, h2

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

4. 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.

5. 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.

6. 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.

7. 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

8. Stationary policy for minimizing total average discounted costs over an infinite horizon This equation can be solved iteratively. The interesting case is a →∞

9. Minimize total average cost over a finite horizon of n steps Solve recurrences in n steps, starting from

10. Steady-state average cost per step, independent of the starting state Obtained by simulation

11. 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)

12. Varying l

13. Varying m

14. Varying x

15. Varying Holding Cost