Finite-Source Multiserver Queue with Preemptive Priorities

Finite-Source Multiserver Queue with Preemptive Priorities

Alexandre BrandwajnSchool of Engineering

University of California, Santa [email protected]

Plan

Motivation System considered Simple recurrent analysis Alternative method Classes on several priority levels Conclusions

Motivation

Priority service computer application

Finite number of request sources Multiple servers Arbitrary number of classes

System considered

Assumptions

M servers c classes exponentially

distributed idle times service demands

1 class / priority level

Assumptions

Class i, i = 1,…c

Ni sources

1/i mean idle time

1/i mean service demand

Class 1 highest priority

Simple recurrent analysis

Simple recurrent analysis

One class at a time state description (ni,li)

ni users of class i

li servers unavailable

servers vanish i(ni,li)

reappear i(ni,li)

Simple recurrent analysis

Approximation i(ni,li) i(li)

i(ni,li) i(li)

two-dimensional birth & death p(ni,li)

starting with class 1

Simple recurrent analysis

i+1(li+1) = l

l

i

i

0

1

p(ni=li+1-li,li)[i(li)+(Ni-ni)i]/ p(li+1),

for li+1 = 0,…,min(M-1,

i

j1

Nj)

Simple recurrent analysis

i+1(li+1) = l

l

i

i

0

1

p(ni=li+1-li,li)[i(li)+s(ni,li)i]/p(li+1)

for li+1 = 1,…,min(M, j

i

1

Nj)

Results with 3 serversSet Solution Method Class Throughput +- Mean Number +-3 exact 1 0.521 0.787

2 7.394 16.303recurrence 1 0.521 0.787

2 7.394 16.303

5 simulation 1 0.524 0.007 0.792 0.0022 14.402 0.067 2.119 0.0103 0.387 0.007 8.452 0.044

recurrence 1 0.521 0.7872 14.409 2.1183 0.389 8.445

6 simulation 1 0.464 0.011 0.452 0.0152 19.527 0.035 1.105 0.0073 1.506 0.024 5.490 0.1054 0.492 0.045 9.434 0.069

recurrence 1 0.455 0.4552 19.488 1.1023 1.527 5.4194 0.440 9.560

Results with 2 servers

Set Solution Method Class Throughput +- Mean Number +-4 exact 1 0.516 0.841

2 10.116 2.977recurrence 1 0.516 0.841

2 10.117 2.977

7 simulation 1 0.274 0.006 0.282 0.0072 11.510 0.039 0.705 0.0033 0.786 0.007 1.627 0.0314 1.209 0.024 2.789 0.0355 0.905 0.023 4.542 0.017

recurrence 1 0.273 0.2742 11.500 0.7003 0.809 1.5734 1.440 2.5605 0.532 4.734

Parameter sets

SetNumber of

ClassesClass

Number of Sources

Mean Source Idle Time

Mean Service Time

1 2 1 6 10 1.52 18 5 3

2 2 1 6 10 1.52 18 0.5 0.3

3 2 1 6 10 1.52 20 0.5 0.3

4 2 1 6 10 1.52 10 0.2 0.1

5 3 1 6 10 1.52 5 0.2 0.13 10 4 2

6 4 1 5 10 12 5 0.2 0.053 10 3 14 10 1 0.01

7 5 1 3 10 12 3 0.2 0.053 4 3 14 4 1 0.25 5 0.5 0.1

Set 5 with 3 servers

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Simulation Recurrence

Class 1Class 2Class 3

Set 6 with 3 servers

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Simulation Recurrence

Class 1Class 2Class 3Class 4

Set 7 with 2 servers

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Simulation Recurrence

Class 1Class 2Class 3Class 4Class 5

Simple recurrent analysis

Generally within confidence intervals occasionally more significant errors

more likely with longer service at higher priority

not systematic growth with ratio of service times

– peak and vanish

Alternative method

Alternative method

Alternative approach

Pair (i, i+1) State description

(ni,ni+1,li)

ni, ni+1 users of class i and i+1

li servers unavailable to pair

Alternative approach

Servers vanishi(ni,li)

reappeari(ni,li)

Pair (1,2) keep

other (i,i+1) keep results for i+1

Set 6 with 3 servers

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Simulation Recurrence Pairwise

Class 1Class 2Class 3Class 4

Set 7 with 2 servers

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Simulation Recurrence Pairwise

Class 1Class 2Class 3Class 4Class 5