finite-source multiserver queue with preemptive priorities
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Finite-Source Multiserver Queue with Preemptive Priorities
Alexandre BrandwajnSchool of Engineering
University of California, Santa Cruzalexb@cse.ucsc.edu
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
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