Finite-Source Multiserver Queue with Preemptive Priorities Alexandre Brandwajn School of Engineering...
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![Page 1: Finite-Source Multiserver Queue with Preemptive Priorities Alexandre Brandwajn School of Engineering University of California, Santa Cruz alexb@cse.ucsc.edu.](https://reader030.fdocuments.net/reader030/viewer/2022032800/56649d4e5503460f94a2da3e/html5/thumbnails/1.jpg)
Finite-Source Multiserver Queue with Preemptive Priorities
Alexandre BrandwajnSchool of Engineering
University of California, Santa [email protected]
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Plan
Motivation System considered Simple recurrent analysis Alternative method Classes on several priority levels Conclusions
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Motivation
Priority service computer application
Finite number of request sources Multiple servers Arbitrary number of classes
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System considered
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Assumptions
M servers c classes exponentially
distributed idle times service demands
1 class / priority level
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Assumptions
Class i, i = 1,…c
Ni sources
1/i mean idle time
1/i mean service demand
Class 1 highest priority
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Simple recurrent analysis
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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)
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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
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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)
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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)
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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
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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
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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
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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
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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
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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
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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
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Alternative method
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Alternative method
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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
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Alternative approach
Servers vanishi(ni,li)
reappeari(ni,li)
Pair (1,2) keep
other (i,i+1) keep results for i+1
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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
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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