Optimal Scheduling Among Intermittently Unavailable

ServersSimon Martin & Isi Mitrani

University of Newcastle upon Tyne

Model

Arrivals

1, 1

policy

2, 2

Parameters

• Arrival rate = At queue i (i=1,2)

• Average service time = 1/i

• Average operative period = 1/i

• Average repair time = 1/i

• Average holding cost = ci

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.

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.

• 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.

Uniformization

222111 ,max,max

1;

max

ijii

ijij

jij

i

rq

rq

r

State of discrete time Markov chainS = (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

Stationary policy for minimizing total average discounted costs over

an infinite horizon

S

dSVSSqdcScSV ,min

This equation can be solved iteratively.

The interesting case is →∞

Minimize total average cost over a finite horizon of n steps

S

nd

n SVSSqdcScSV 1,min

Solve recurrences in n steps, starting from

00 SV

Steady-state average cost per step, independent of the starting state

n

SVV n

n lim

Obtained by simulation

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)

Varying

0

10

20

30

40

50

60

70

80

90

100

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Lambda

Ave

rag

e C

ost

Random

Selective(Nigel)

ShortestQueue

Heuristic

Optimal

Varying

0

50

100

150

200

250

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Mu1

Ave

rag

e C

ost

Random

Selective(Nigel)

ShortestQueue

Heuristic

Optimal

Varying

0

20

40

60

80

100

120

140

160

180

200

0 0.2 0.4 0.6 0.8 1 1.2

Xi1

Av

era

ge

Co

st

Random

Selective(Nigel)

ShortestQueue

Heuristic

Optimal

Varying Holding Cost

0

20

40

60

80

100

120

140

0 0.5 1 1.5 2 2.5 3 3.5 4

C1

Ave

rag

e co

st

Random

Selective(Nigel)

ShortestQueue

Heuristic

Optimal