optimal routing in general finite multi-server queueing networks · 1. the optimal topology problem...
TRANSCRIPT
Optimal routing in general finite multi-server queueingnetworksCitation for published version (APA):Woensel, van, T., & Cruz, F. R. B. (2014). Optimal routing in general finite multi-server queueing networks. PLoSONE, 9(7), e102075-1/15. [e102075]. https://doi.org/10.1371/journal.pone.0102075
DOI:10.1371/journal.pone.0102075
Document status and date:Published: 01/01/2014
Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publication
General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverne
Take down policyIf you believe that this document breaches copyright please contact us at:[email protected] details and we will investigate your claim.
Download date: 09. Sep. 2021
Optimal Routing in General Finite Multi-Server QueueingNetworksTom van Woensel1*, Frederico R. B. Cruz2
1 Department of Industrial Engineering & Innovation Sciences, Technische Universiteit Eindhoven, Eindhoven, The Netherlands, 2 Departamento de Estatıstica,
Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil
Abstract
The design of general finite multi-server queueing networks is a challenging problem that arises in many real-life situations,including computer networks, manufacturing systems, and telecommunication networks. In this paper, we examine theoptimal routing problem in arbitrary configured acyclic queueing networks. The performance of the finite queueingnetwork is evaluated with a known approximate performance evaluation method and the optimization is done by means ofa heuristics based on the Powell algorithm. The proposed methodology is then applied to determine the optimal routingprobability vector that maximizes the throughput of the queueing network. We show numerical results for some networksto quantify the quality of the routing vector approximations obtained.
Citation: van Woensel T, Cruz FRB (2014) Optimal Routing in General Finite Multi-Server Queueing Networks. PLoS ONE 9(7): e102075. doi:10.1371/journal.pone.0102075
Editor: Gerardo Adesso, University of Nottingham, United Kingdom
Received April 15, 2014; Accepted June 13, 2014; Published July 10, 2014
Copyright: � 2014 van Woensel, Cruz. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permitsunrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The authors confirm that all data underlying the findings are fully available without restriction. All relevant data are within the paper and itsSupporting Information files.
Funding: FRBC was supported by the Conselho Nacional de Desenvolvimento Cientıfico e Tecnologico (CNPq) of the Ministry for Science and Technology ofBrazil, grant number 303388/2010-2, and Fundacao de Amparo a Pesquisa do Estado de Minas Gerais (FAPEMIG), grant number CEX-PPM-00071-12. The fundershad no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* Email: [email protected]
Introduction
The design of networks with random arrivals, random service
times, multiple servers, and a finite number of buffer spaces is a
challenging problem that arises in many real-life situations, e.g. in
manufacturing systems [1,2], computer networks [3,4], public
services [5,6], call centers [7,8], pedestrian and vehicular traffic
[9–12], among other situations. This problem is challenging
because finite queueing networks are notoriously difficult to
analyze analytically, and closed form expressions are not easily
constructed for the performance measures of such systems. Also
note that the underlying network design problems involved are
usually very hard to solve.
In fact, there are several distinct network design optimization
problems associated with finite queueing networks. According to
Daskalaki & Smith [13] the optimal network design problem can
be split up into three interrelated optimization problems:
1. The optimal topology problem (OTOP), which deals with
decisions of the design of the network itself, that is, the number
of nodes (e.g. workstations, warehouses, delivery points, etc.)and arcs (e.g. corridors, conveyors, escalators, etc.) and the
general configuration of these two components;
2. The optimal routing problem (OROP), which deals with
determining the routing probabilities in the network defined by
the first problem;
3. The optimal resource allocation problem (ORAP), which deals
with the optimal allocation of the scarce resources in the
network, e.g. the number of buffers (i.e., the buffer allocation
problem, BAP) and the number of servers (i.e., the server
allocation problem, CAP).
These three problems are challenging and difficult optimization
problems. For an arbitrary topology, the OTOP is shown to be
NP{hard [14], and the same is conjectured for the general
ORAP [15].
Previous work focused mainly on the ORAP in open finite
acyclic queueing network settings. Both BAP and CAP are
probably among the most well-known optimal resource allocation
problems [16]. For instance, Cruz et al. [17] and Smith et al. [18]
looked into the BAP, both in a single and in a multi-server setting,
and Smith et al. [19] proposed algorithms to solve the CAP.
However, the routing probabilities are usually assumed to be
known beforehand for BAP and CAP [20].
The overall research objective of this paper is to build a relevant
model and solution methodology for the system’s throughput
maximization problem. In this paper, we focus on optimizing the
routing probabilities through the queueing network, i.e. the
OROP. A similar research question is tackled by Daskalaki &
Smith [13] in which they evaluated the joint effect of buffer
allocation and routing on the throughput. Earlier, Gosavi & Smith
[21] focused on the routing optimization problem related to the
overall objective of throughput maximization. The common
ground of both papers is that they used queueing networks with
single servers and exponential service times [13,21]. Kerbache &
Smith [22] considered, for different product classes, the optimal
routes conjoint with a variant of the optimal topology problem to
determine the connected arcs in a single server queueing network
setting. Distinct from Daskalaki & Smith [13] and Gosavi & Smith
PLOS ONE | www.plosone.org 1 July 2014 | Volume 9 | Issue 7 | e102075
[21] is that Kerbache & Smith [22] considered general arrival
times, general service times, and single server queues. Secondly,
Gosavi & Smith [21] did not consider the general expansion
method (GEM) in their analysis as the evaluation tool [23].
Specifically, we examine the OROP, by means of a combina-
tion of the GEM and a heuristic based on the Powell algorithm
[24], specifically for acyclic networks of M=G=c=K queues, which
in Kendall notation means a queueing system with Markovian
arrival rates, General service times, c parallel servers, and a total
capacity of K items, including those items in service (practical
applications to M=G=c=K queueing networks include manufac-
turing and service systems [25] and transportation and material
handling systems [26]). The results are compared to simulations to
attest for the quality of the routing vectors obtained. Besides,
another important contribution of this paper is to present helpful
approximations to swift managerial decisions regarding the
optimal routing probability vectors to maximize the overall
throughput in a network of finite general-service queues. We also
present important empirical arguments to show that these
approximations are effective.
This paper is organized as follows. The next section describes in
detail the mathematical model formulation considered and
elaborates on both the performance evaluation tool and on the
optimization procedure. In the following section detailed results
are given for the problem on hand. Finally, the last section
concludes this paper with final remarks and topics for future work
in the area.
Materials and Methods
Mathematical Programing FormulationThe problem is defined on a digraph D~(V ,A), in which V is
the set of vertexes (finite queues) and A is the set of arc
(connections between the queues). Each vertex (queue) is
characterized by Poisson arrivals, general service, and multiple
servers. Mathematically, the optimal routing problem can be
formulated as follows.
(OROP):
maxH(a ), ð1Þ
subject to:
0ƒai, jƒ1, V (i, j) [ A, ð2Þ
XVj[d(i)
ai, j~1, V i [ V , ð3Þ
in which H(a ) is the throughput, which is the objective that must
be maximized, a the optimal routing probability matrix, a:½a i, j �,i.e. the matrix that maximizes the objective function of this
predefined network, and d(i) is the set of succeeding vertexes of
vertex i, that is, d(i):fjj (i, j) [ Ag:The throughput will thus be affected by the effective routing of
jobs through the network, the variability of the service distribution,
the number of servers, and the number of buffers. It should be
clear that the above model is a highly difficult stochastic
programming problem to handle due to the inherent non-linearity
of the objective function combined with the absence of any closed-
form expressions for the throughput H(a):
Proposed AlgorithmFigure 1 presents the proposed algorithm to solve the OROP in
order to provide more insights into the interaction between the
performance evaluation tool and the optimization method.
The initial routing probability vector is conveniently set to the
inverse of the number of nodes after a split,
a (init)i, j ~
1
ni
, V(i, j) [ A, ð4Þ
in which ni is the number of succeeding nodes to node i, that is,
the cardinality of set D(i): The optimization step itself is an
iteration in which new solutions are generated following the Powell
logic until convergence, that is, until the difference in H,
DH:(H(k){H(k{1)), is less than a predefined tolerance e:The Powell algorithm can be described as an unconstrained
optimization procedure that does not require the calculation of
first derivatives of the function. Numerical examples have shown
that the method is capable of minimizing a function with up to
twenty variables [24,27]. The Powell method locates the minimum
of a non-linear function f (x) by successive uni-dimensional
searches from an initial starting point x(k) along a set of conjugate
directions. These conjugate directions are generated within the
procedure itself. The Powell method is based on the idea that if a
minimum of a non-linear function f (x) is found along p conjugate
directions in a stage of the search, and an appropriate step is made
in each direction, the overall step from the beginning to the p-th
step is conjugate to all of the p sub-directions of the search. We
have seen remarkable success in the past with coupling Powell
algorithm and the GEM [19]. We discuss the GEM in detail now,
which is also the method we used to obtain the performance
measures for the problem studied in this paper.
Performance EvaluationIn previous papers (see e.g. Kerbache & Smith [23,28]) the GEM
has been successfully used to evaluate the performance measures of
acyclic networks of finite queues. The GEM is a robust and effective
approximation technique that is basically a combination of repeated
trials and node-by-node decomposition in which each queue is
analyzed separately and then corrections are made in order to take
into account the interrelation between the queues in the network.
Figure 1. Structured overview of the methodology.doi:10.1371/journal.pone.0102075.g001
Optimal Routing
PLOS ONE | www.plosone.org 2 July 2014 | Volume 9 | Issue 7 | e102075
The GEM has three stages, Network Reconfiguration, ParameterEstimation, and Feedback Elimination, to be described as follows.
Stage I: Network Reconfiguration. The first step in the
GEM involves reconfiguring the network. An artificial vertex hj is
added preceding each finite vertex j in the network. The artificial
vertex is added to register the blocked customers at node j and is
modeled as an M=G=? queue, as shown in Figure 2.
When an entity leaves vertex i, vertex j may be blocked with
probability pKj, or unblocked, with probability (1{pKj
): Under
blocking, the entities are rerouted to vertex hj for a delay while
node j is busy. Vertex hj helps to accumulate the time an entity has
to wait before entering vertex j and to compute the effective arrival
rate to vertex j: In other words, the GEM ultimate goal is to
provide an approximation scheme to update the service rates at the
upstream vertex i to take into account all blocking after service
caused by the downstream vertex j :
~mm{1i ~m{1
i zpKj(m’hj
){1, ð5Þ
in which m i is the exponential service rate at vertex i, pKjis the
blocking probability of finite queue j of size Kj , m’hjis the corrected
exponential service rate at the artificial vertex hj , and ~mmi is the
updated service rate at vertex i: As a final note, an important point
about this process is that we do not physically modify the networks,
only represent the expansion process through the software.
Stage II: Parameter Estimation. In the second stage, the
parameters pK , p’K , and m h must be estimated, which is done
essentially by utilizing known results for queueing theory. Index j is
omitted for simplicity.
pK : Analytical results from the M=M=c=K queue provide
the following expression for the blocking probability pK :
pK~1
cK{cc!
�
m
� �K
p0, ð6Þ
in which
p0~Xc{1
n~0
1
n!
�
m
� �n
z(�=m)c
c!
1{ �=(cm)½ �K{cz1
1{�=(cm)
" #{1
, ð7Þ
for �=(cm)=1:
However, the interest is on M=G=c=K models, for
which there is not exact closed form expression for pK :Then approximations must be used and Kimura’s [29]
two moment approximation has proven to be very
effective in similar cases [25,30]. For example, let us fix
c~2 and the following closed form expression can be
developed from Equation (6), for the optimal buffer size
BM~K{2 for Markovian M=M=2=K queues, as a
function of the blocking probability:
BM~
ln1
2
pK 2mz�ð Þ2m{�zpK�
� �ln (r)
{2: ð8Þ
The following Kimura’s two moment approximation can
be used to approximate the optimal buffer size B (s2) of
a general service M=G=2=K queue:
B (s2)~BMz NINTs2{1
2
ffiffiffirp
BM
� �, ð9Þ
in which s2 is the squared coefficient of variation of the
service time distribution at the queue, r:�=(cm) is the
traffic intensity, BM is the exact buffer size for a
respective Markovian system, and NINT( x) is the
nearest integer to x: Now, if we invert Equation (9) to
Figure 3. Basic split network B1.doi:10.1371/journal.pone.0102075.g003
Figure 2. The generalized expansion method.doi:10.1371/journal.pone.0102075.g002
Optimal Routing
PLOS ONE | www.plosone.org 3 July 2014 | Volume 9 | Issue 7 | e102075
solve for pK we achieve:
pK~2r
2
ffiffire
ps2{
ffiffire
pzK
2z
ffiffire
ps2{
ffiffire
p !
(2m{�)
{2r
2
ffiffire
ps2{
ffiffire
pzK
2z
ffiffire
ps2{
ffiffire
p !
�z2mz�
: ð10Þ
This is a process that can be extended for cw2: In fact,
expressions for pK of up to c~500 are available [30].
Another expressions, for c~3, . . . ,10, are included in
the software so that we have a complete set of blocking
probabilities for c [ f1, . . . ,10g:p’K : Since there is no closed form solution for this quantity the
following approximation is used given by Labetoulle &
Pujolle [31] obtained using diffusion techniques.
Figure 4. The shape of the objective function.doi:10.1371/journal.pone.0102075.g004
Figure 5. Basic split network B2.doi:10.1371/journal.pone.0102075.g005
Optimal Routing
PLOS ONE | www.plosone.org 4 July 2014 | Volume 9 | Issue 7 | e102075
Ta
ble
1.
Re
sult
sfo
rtw
o-b
ran
chsp
litn
etw
ork
s. �~
3�~
5�~
7
Se
tm
18
m2
:m
3s2 1 :
s2 3
8s2 2
Ha
1,2
a1,3
Ha
1,2
a1,3
Ha
1,2
a1,3
B1
a28
1:
21:0
80:5
:0:5
2.9
16
50
.33
37
0.6
66
34
.29
50
0.3
33
40
.66
66
5.0
93
00
.33
34
0.6
66
6
B1
b1:0
81:0
:1:0
2.9
00
40
.33
34
0.6
66
64
.21
93
0.3
33
40
.66
66
4.9
81
90
.33
34
0.6
66
6
B1
c1:0
81:5
:1:5
2.8
85
00
.33
37
0.6
66
34
.29
50
0.3
33
40
.66
66
4.8
88
80
.33
34
0.6
66
6
B1
d28
2:
21:0
80:5
:0:5
2.9
73
30
.49
89
0.5
01
14
.65
45
0.5
00
00
.50
00
5.8
77
70
.50
00
0.5
00
0
B1
e1:0
81:0
:1:0
2.9
67
20
.49
86
0.5
01
44
.60
10
0.5
00
00
.50
00
5.7
69
40
.50
00
0.5
00
0
B1
f1:0
81:5
:1:5
2.9
60
90
.49
84
0.5
01
64
.55
25
0.5
00
00
.50
00
5.6
76
50
.50
00
0.5
00
0
B1
g28
3:
21:0
80:5
:0:5
2.9
87
80
.59
93
0.4
00
74
.88
07
0.6
00
00
.40
00
6.3
17
60
.60
00
0.4
00
0
B1
h1:0
81:0
:1:0
2.9
84
70
.59
93
0.4
00
74
.85
68
0.6
00
00
.40
00
6.2
30
50
.60
01
0.3
99
9
B1
i1:0
81:5
:1:5
2.9
81
60
.59
90
0.4
01
04
.83
37
0.6
00
00
.40
00
6.1
52
90
.59
98
0.4
00
2
B1
j28
1:
11:0
80:0
:1:0
2.6
74
30
.52
22
0.4
77
83
.50
07
0.5
03
90
.49
61
3.7
98
10
.48
43
0.5
16
7
B1
k1:0
80:5
:1:0
2.6
48
90
.51
00
0.4
90
03
.45
44
0.5
02
10
.49
79
3.7
57
30
.49
44
0.5
05
6
B1
l1:0
81:0
:1:0
2.6
27
80
.50
00
0.5
00
03
.41
61
0.5
00
00
.50
00
3.7
21
20
.50
00
0.5
00
0
B1
m1:0
81:5
:1:0
2.6
10
10
.49
18
0.5
08
23
.38
39
0.4
97
80
.50
22
3.6
89
80
.50
35
0.4
96
5
B1
n1:0
82:0
:1:0
2.5
94
70
.48
50
0.5
15
03
.35
67
0.4
96
00
.50
40
3.6
62
60
.50
59
0.4
94
1
B1
o28
2:
21:0
80:0
:1:0
2.9
73
90
.53
09
0.4
69
04
.66
12
0.5
26
50
.47
35
5.8
91
00
.51
81
0.4
81
9
B1
p1:0
80:5
:1:0
2.9
70
40
.51
32
0.4
86
84
.62
86
0.5
11
60
.48
84
5.8
24
20
.50
81
0.4
91
9
B1
q1:0
81:0
:1:0
2.9
67
10
.49
86
0.5
01
44
.60
10
0.4
99
90
.50
01
5.7
69
40
.50
00
0.5
00
0
B1
r1:0
81:5
:1:0
2.9
64
20
.48
64
0.5
13
64
.57
73
0.4
90
30
.50
97
5.7
23
40
.49
33
0.5
06
7
B1
s1:0
82:0
:1:0
2.9
61
50
.47
58
0.5
24
24
.55
67
0.4
82
30
.51
77
5.6
84
20
.48
77
0.5
12
3
B1
t28
3:
31:0
80:0
:1:0
2.9
93
80
.53
41
0.4
65
94
.93
72
0.5
28
30
.47
17
6.6
04
70
.56
90
0.4
31
0
B1
u1:0
80:5
:1:0
2.9
92
90
.51
33
0.4
84
74
.92
89
0.5
10
70
.48
93
6.5
31
90
.51
25
0.4
87
5
B1
v1:0
81:0
:1:0
2.9
92
00
.49
95
0.5
00
54
.92
14
0.4
96
50
.50
35
6.5
03
90
.43
14
0.5
68
6
B1
w1:0
81:5
:1:0
2.9
91
20
.48
57
0.5
14
34
.91
47
0.4
84
40
.51
56
6.4
89
10
.43
14
0.5
68
6
B1
x1:0
82:0
:1:0
2.9
90
50
.47
42
0.5
25
84
.90
86
0.4
74
10
.52
59
6.4
74
90
.43
14
0.5
68
6
do
i:10
.13
71
/jo
urn
al.p
on
e.0
10
20
75
.t0
01
Optimal Routing
PLOS ONE | www.plosone.org 5 July 2014 | Volume 9 | Issue 7 | e102075
p’K~mjzmh
mh
{� (rK
2 {rK1 ){(rK{1
2 {rK{11 )
� �mh (rKz1
2 {rKz11 ){(rK
2 {rK1 )
� � !{1
,ð11Þ
in which r1 and r2 are the roots to the polynomial
�{(�zmhzmj)xzmhx2~0,
with �~�j{�h(1{p’K ), in which �j and �h are the
actual arrival rates to the finite and artificial holding
nodes respectively. Furthermore, the arrival rate to the
finite node j is given by:
�j~ ~��i(1{pK )~~��i{�h:
mh : The delay distribution of a blocked customer at the
holding node has the same distribution as the remaining
service time of the customer being serviced at the node
doing the blocking. Using renewal theory, one can show
that the remaining service time distribution has the
following rate mh :
mh~2m
1zs2m2, ð12Þ
in which s2 is the service time variance given by
Kleinrock [32]. Notice that if the service time distribu-
tion at the finite queue doing the blocking is exponential
with rate mj , then:
mh~mj ,
that is, the service time at the artificial node is also
exponentially distributed with rate mj : When the service
time of the blocking node is not exponential, then mh will
be affected by s2:Stage III: Feedback Elimination. This stage is simply to
eliminate the feedback loop, by recomputing the service time at
vertex hk: The updated service rate is given by:
m’h~(1{p’K )mh:
Summary. Similar equations can be established with respect
to each of the finite vertexes (queues). Ultimately, we have
simultaneous non-linear equations in variables pK , p’K , and mh,
along with auxiliary variables such as mj and ~��i: Solving these
equations simultaneously, we can compute all performance
measures of the network.
Numerical Results and Discussion
The software implementation is currently in Fortran 77. The
compiler used was the DIGITAL Visual Fortran, Version 6, with
the IMSL Fortran 90 MP Library version 3.0 for Microsoft
Windows, to solve the nonlinear equations from the GEM. In our
implementation, we set e~10{1,000, which proved to be effective
based on the experiments. We first discuss the shape of the
objective function. Secondly, we will give more insights for a
number of split structures. We end the numerical results with some
Ta
ble
2.
Pe
rtu
rbat
ion
sar
ou
nd
the
op
tim
also
luti
on
of
two
-bra
nch
split
ne
two
rks.
Se
tB
1e
(ba
lan
ced
)S
et
B1
b(u
nb
ala
nce
d)
a1,2
a1,3
�~
3�~
5�~
7�~
3�~
5�~
7
0.1
00
.90
2.7
38
83
.78
15
4.3
37
02
.73
86
3.7
80
34
.33
14
0.2
00
.80
2.8
27
54
.10
84
4.9
12
32
.82
47
4.0
82
14
.77
97
0.3
00
.70
2.8
93
54
.37
93
5.3
67
22
.87
69
4.2
09
94
.97
06
0.3
30
.66
……
…2
.90
04
4.2
19
34
.98
19
0.4
00
.60
2.9
58
24
.56
97
5.6
66
52
.87
77
4.1
83
44
.94
12
0.4
50
.55
2.9
64
94
.58
58
5.7
43
5…
……
0.5
00
.50
2.9
67
24
.60
10
5.7
69
32
.79
74
4.0
08
64
.74
52
0.5
50
.45
2.9
64
74
.58
58
5.7
43
5…
……
0.6
00
.40
2.9
58
24
.56
97
5.6
66
52
.66
53
3.7
44
24
.40
38
0.7
00
.30
2.8
93
54
.37
94
5.3
67
22
.49
04
3.3
45
93
.92
49
0.8
00
.20
2.8
27
54
.10
84
4.9
12
42
.28
28
2.8
98
93
.33
74
0.9
00
.10
2.7
38
83
.78
15
4.3
37
02
.05
14
2.4
27
43
.12
35
do
i:10
.13
71
/jo
urn
al.p
on
e.0
10
20
75
.t0
02
Optimal Routing
PLOS ONE | www.plosone.org 6 July 2014 | Volume 9 | Issue 7 | e102075
Ta
ble
3.
Re
sult
sfo
rth
ree
-bra
nch
split
ne
two
rks.
�~
3�~
5�~
7
Se
tm
18
m2
:m
4?
m3
s2 1
8s2 2
:s2 4
?s2 3
Ha
1,2
a1,3
a1,4
Ha
1,2
a1,3
a1,4
Ha
1,2
a1,3
a1,4
B2
a28
2:
2?
21:0
80:0
:1:0?
1:0
2.9
93
30
.37
58
0.3
12
00
.31
22
4.9
32
60
.35
87
0.3
20
40
.32
09
6.5
64
20
.38
45
0.3
33
30
.28
22
B2
b1:0
80:5
:1:0?
1:0
2.9
92
60
.34
69
0.3
26
50
.32
65
4.9
26
80
.34
31
0.3
28
30
.32
87
6.5
20
90
.34
45
0.3
27
80
.32
77
B2
c1:0
81:0
:1:0?
1:0
2.9
92
10
.33
30
0.3
33
50
.33
35
4.9
21
70
.33
04
0.3
34
60
.33
50
6.4
98
20
.33
33
0.3
33
30
.33
33
B2
d1:0
81:5
:1:0?
1:0
2.9
91
50
.32
10
0.3
39
50
.33
95
4.9
17
40
.31
99
0.3
39
90
.34
01
6.4
97
30
.28
66
0.3
58
00
.35
54
B2
e1:0
82:0
:1:0?
1:0
2.9
91
00
.31
09
0.3
44
50
.34
46
4.9
13
50
.31
09
0.3
44
40
.34
46
6.4
62
30
.31
65
0.3
41
70
.34
18
B2
f28
1:
3?
21:0
80:0
:1:0?
1:0
2.9
92
70
.18
60
0.3
25
70
.48
83
4.9
27
70
.18
21
0.3
26
70
.49
12
6.5
23
90
.18
24
0.3
27
00
.49
06
B2
g1:0
80:5
:1:0?
1:0
2.9
92
30
.17
42
0.3
30
80
.49
49
4.9
24
60
.17
17
0.3
31
00
.49
73
6.5
09
70
.17
37
0.3
30
50
.49
57
B2
h1:0
81:0
:1:0?
1:0
2.9
92
10
.16
55
0.3
34
00
.50
05
4.9
22
10
.16
42
0.3
34
00
.50
18
6.5
03
40
.20
54
0.3
63
40
.43
11
B2
i1:0
81:5
:1:0?
1:0
2.9
91
80
.15
82
0.3
36
80
.50
50
4.9
19
90
.15
81
0.3
36
50
.50
53
6.4
88
60
.16
09
0.3
35
60
.50
35
B2
j1:0
82:0
:1:0?
1:0
2.9
91
60
.15
22
0.3
39
10
.50
87
4.9
18
10
.15
23
0.3
38
90
.50
88
6.4
80
60
.15
62
0.3
37
50
.50
63
�~
3�~
5�~
7
Se
tc
18
c2
:c
4?
c3
?s2 3
Ha
12
a1
3a
14
Ha
12
a1
3a
14
Ha
12
a1
3a
14
B2
k28
1:
3?
21:0
80:0
:1:0?
1:0
2.9
94
50
.16
70
0.3
04
90
.53
44
4.9
27
70
.19
26
0.3
29
40
.47
80
6.6
39
10
.20
71
0.2
86
80
.50
62
B2
l1:0
80:5
:1:0?
1:0
2.9
93
90
.13
82
0.3
12
60
.54
92
4.9
19
50
.16
79
0.3
40
70
.19
14
6.6
02
80
.18
66
0.2
87
50
.52
59
B2
m1:0
81:0
:1:0?
1:0
2.9
93
50
.12
23
0.3
19
10
.55
87
4.9
13
80
.15
07
0.3
48
30
.50
09
6.5
55
30
.16
37
0.3
22
40
.51
38
B2
n1:0
81:5
:1:0?
1:0
2.9
93
20
.11
38
0.3
23
10
.56
31
4.9
09
40
.13
91
0.3
53
50
.50
74
6.5
38
20
.15
26
0.3
27
00
.52
05
B2
o1:0
82:0
:1:0?
1:0
2.9
92
90
.10
12
0.3
28
30
.57
05
4.9
05
90
.13
01
0.3
57
30
.51
26
6.5
24
80
.14
39
0.3
30
50
.52
56
do
i:10
.13
71
/jo
urn
al.p
on
e.0
10
20
75
.t0
03
Optimal Routing
PLOS ONE | www.plosone.org 7 July 2014 | Volume 9 | Issue 7 | e102075
s2 1
8s2 2
:s2 4
complex network structures. Please bear in mind that the range of
possible experiments is exponential itself, so we have determined a
very selected, but representative sample to present.
Shape of the Objective FunctionIt is interesting to analyze the shape of the objective function for
the optimization problem described earlier. The case discussed
here is defined as follows. We have a three node network with a
split into two branches, as seen in Figure 3. The general
parameters for the base case are c1~4, K1~20, and ci~2 and
Ki~2, for i~2,3: The criteria to select those parameters is such
that the number of servers c1 and the total capacity K1 of node 1 is
larger than the others as to prevent it from becoming a bottleneck.
We are particularly interested in the relationship between the
overall throughput H~h1zh2, the routing probability a 1, 2, the
arrival rate �1, and the squared coefficient of variation of node 2,
s22: Consequently, we set mi~2, for all nodes, and s2
1~s23~1: The
sensitivity of these settings on the throughput is not analyzed now,
but is amongst others the subject of study in the next sections.
Next, we enumerate all possible combinations for �1, a 1, 2, and s22,
and then analytically obtain the corresponding throughput H,which is shown in Figure 4 (always on the vertical axis), as a
function of �1, a 1, 2, and s22:
Figure 4-(a) clearly shows that the arrival rate is interacting with
the routing probability. For low arrival rates, the network has low
utilization. Consequently, different routing probabilities do not
result in large changes in throughput H: On the other hand, for
large arrival rates, �1w5, one clearly sees an optimal point in
regard to the routing probability. Due to the symmetrical structure
considered, a 50% split seems to be optimal here. Figure 4-(b)
looks into the joint effect of changing the squared coefficient of
variation, s22, together with the routing probability a 1, 2: Again the
inverted U-shape effect with a maximum around the 50% routing
probability is visible. Next to this, it is clear that increasing the
squared coefficient of variation from 0 to 2 reduces the overall
throughput H but has a smaller impact on throughput than the
routing probability. Based on this simple network structure, it is
evident that the routing probabilities and the squared coefficient of
variation affect the throughput to a large extent. Consequently,
correctly setting the routing matrix a leads to significant gains in
terms of throughput.
Basic Split NetworksIn this section, we analyze further the two-branch network from
Figure 3 and include in our analysis the three-branch network seen
in Figure 5. We are interested in assessing the influence of the
number of servers ci, total capacities Ki, service rates mi, and
squared coefficient of variation of the service times s2i , Vi [ V , in
the model OROP, Equations (1) – (3). We choose to start with
these two variants of a basic split structure as, from a routing
allocation point of view, splits are the key building blocks in a
generally configured network. The nodes after the splits are the
ones of interest here. The first buffer K1~20 is larger than the
others (Ki~2, i~2,3,4) as this will help to prevent the first queue
of becoming a bottleneck node. The arrival rate �1 is set equal to
values f3,5,7g:Split with Two Branches. The top part of Table 1 gives the
results for a two-branch split networks for unbalanced service rates
m and different squared coefficients of variation s22 and s2
3: In these
cases the service rate of node 2 is made either relatively lower
(m2~1 versus m3~2), or equal (m2~2 versus m3~2), or higher
(m2~3 versus m3~2) than the service rate of node 3. The base
cases (sets B1d to B1f) show that the routing probability is roughly
equal to 0.5 when the nodes after the split are identical (that is,
same number of servers, capacities, service rates, and squared
coefficient of variation). Moreover, this results appears to be
insensitive to changes in the squared coefficient of variation of
both nodes after the split. Of course, the throughput H is affected
(reduced) due to the changes (increase) in the variability. Secondly,
changing the service rate of node 2, m2 (and keeping all other
parameters equal to the base case settings), clearly shows that the
fast nodes receive preference over the slow nodes. For example, in
sets B1a to B1c (i.e., when node 2 is slower than node 3) a lower
routing probability is set to node 2 (0.3334) than the one to node 3
(0.6666).
Rather than changing the squared coefficient of variation of
both nodes after the split, we evaluated some unbalanced cases
where only node 2 is affected by a different squared coefficient of
variation, s22~f0:0,0:5,1:0,1:5,2:0g (sets B1j to B1x, Table 1),
combined with (m1,m2,m3)~f(2,1,1),(2,2,2),(2,3,3)g: For these
cases, we observe that the unbalance caused by the squared
coefficient of variation only slightly changes the routing probability
compared to sets with equal squared coefficients of variation (sets
B1l, B1q, and B1v, Table 1). This is a confirmation of what we
observed when evaluating the objective function earlier in the
previous section. As we are now focusing on the small scale
networks, this conclusion does not mean that the squared
coefficient of variation has little effect in general. It is interesting
to see that the throughput value is reducing as the squared
coefficient of variation goes up although the routing probability is
changing to protect more against the uncertainty in the second
node. This is more prevalent in highly loaded systems.
For the two-branch split networks, we evaluated a number of
routing vectors around the optimal routing obtained. Table 2
shows that the algorithm seems to have reached the optimal
allocation for the routing probabilities into nodes 2 and 3 (set B1e,
Table 1). Of course, one might argue that the optimization is
rather easy due to the symmetric setting of the parameters.
Therefore, we did the same analysis for the same parameter
settings but with a network with unbalance in the service rates (set
B1b, Table 1), also seen in Table 2.
In conclusion, we observed that in previous results the
optimization algorithm tries to balance out the flow taking into
Figure 6. Network structure C1.doi:10.1371/journal.pone.0102075.g006
Optimal Routing
PLOS ONE | www.plosone.org 8 July 2014 | Volume 9 | Issue 7 | e102075
Ta
ble
4.
Re
sult
sfo
rn
etw
ork
stru
ctu
reC
1.
�~
5�~
7
c,s2
Ha
1,2
a1,3
a2,4
a2,5
a3,6
a3,7
Ha
1,2
a1,3
a2,4
a2,5
a3,6
a3,7
C1
ac~
(5,2
,2,2
,2,2
,2)
s~(1
,1,1
,1,1
,1,1
)4
.58
10
0.5
00
00
.50
00
0.4
99
50
.50
05
0.5
00
00
.50
00
5.7
14
20
.50
00
0.5
00
00
.49
86
0.5
01
40
.49
99
0.5
00
1
C1
bc~
(5,2
,2,1
,1,5
,5)
s~(1
,1,1
,1,1
,1,1
)4
.49
78
0.4
00
00
.60
00
0.4
92
40
.50
76
0.5
00
00
.50
00
6.0
64
50
.44
84
0.5
51
60
.48
33
0.5
16
70
.50
00
0.5
00
0
C1
cc~
(5,2
,2,5
,1,5
,1)
s~(1
,1,1
,1,1
,1,1
)4
.60
02
0.5
00
00
.50
00
0.9
25
10
.07
49
0.9
23
90
.07
61
6.4
13
90
.50
00
0.5
00
00
.90
13
0.0
98
70
.90
13
0.0
98
7
C1
dc~
(5,2
,2,2
,2,2
,2)
s~(1
,1,1
,0,0
,2,2
)4
.58
08
0.5
04
20
.49
58
0.4
99
20
.50
08
0.4
99
90
.50
01
6.3
33
50
.50
47
0.4
95
30
.49
93
0.5
00
70
.49
96
0.5
00
4
C1
ec~
(5,2
,2,2
,2,2
,2)
s~(1
,1,1
,0,2
,0,2
)4
.58
26
0.5
00
00
.50
00
0.5
57
70
.44
23
0.5
58
10
.44
19
6.3
39
50
.49
99
0.5
00
10
.44
52
0.5
54
80
.44
64
0.5
53
6
�~
5�~
7
m,s
2H
a1
,2a
1,3
a2,4
a2,5
a3,6
a3,7
Ha
1,2
a1,3
a2,4
a2,5
a3,6
a3,7
C1
fm~
(2,2
,2,1
,1,5
,5)
s~(1
,1,1
,1,1
,1,1
)4
.51
93
0.4
00
00
.60
00
0.4
98
10
.50
19
0.4
99
10
.50
09
6.0
14
80
.43
88
0.5
61
20
.50
00
0.5
00
00
.50
00
0.5
00
0
C1
gm~
(2,2
,2,1
,5,1
,5)
s~(1
,1,1
,1,1
,1,1
)4
.59
64
0.5
00
00
.50
00
0.1
66
50
.83
35
0.1
66
50
.83
35
6.3
99
40
.50
00
0.5
00
00
.16
59
0.8
34
10
.16
64
0.8
33
6
�~
5�~
7
c,m
Ha
1,2
a1,3
a2,4
a2,5
a3,6
a3,7
Ha
1,2
a1,3
a2,4
a2,5
a3,6
a3,7
C1
hc~
(2,2
,2,1
,2,3
,4)
m~
(2,2
,2,1
,5,1
,5)
4.5
97
50
.49
78
0.5
02
20
.06
84
0.9
31
60
.10
87
0.8
91
35
.78
17
0.4
97
50
.50
25
0.0
73
30
.92
67
0.0
99
80
.90
02
C1
ic~
(2,2
,2,4
,2,2
,4)
m~
(2,2
,2,1
,5,1
,5)
4.6
00
30
.49
95
0.5
00
50
.37
24
0.6
27
60
.05
00
0.9
50
05
.78
92
0.4
99
30
.50
07
0.3
45
50
.65
45
0.0
52
50
.94
75
do
i:10
.13
71
/jo
urn
al.p
on
e.0
10
20
75
.t0
04
Optimal Routing
PLOS ONE | www.plosone.org 9 July 2014 | Volume 9 | Issue 7 | e102075
mm
mmm
Ta
ble
5.
Re
sult
sfo
rn
etw
ork
stru
ctu
reC
2.
�~
5�~
7
Ha
2,3
a2,4
a4,5
a4,6
Ha
2,3
a2,4
a4,5
a4,6
C2
ac~
(5,2
,...
,2)
s~(1
,1,.
..,1
)3
.17
19
0.4
00
00
.60
00
0.2
40
40
.75
96
3.3
50
20
.44
64
0.5
53
60
.35
00
0.6
50
0
C2
bc~
(5,2
,...
,2)
s~(1
,1,.
..,1
),c 1
0~
12
.96
99
0.4
00
00
.60
00
0.0
40
90
.95
91
3.0
73
50
.28
57
0.7
14
30
.02
56
0.9
74
4
C2
cc~
(5,2
,...
,2)
s~(1
,1,.
..,1
),c 1
0~
43
.26
60
0.4
00
00
.60
00
0.4
85
80
.51
42
3.4
89
80
.44
76
0.5
52
40
.46
79
0.5
32
1
C2
dc~
(5,2
,...
,2)
s~(1
,1,.
..,1
),m
10~
12
.92
51
0.3
98
80
.60
12
0.2
40
40
.75
96
3.0
85
00
.42
92
0.5
70
80
.00
02
0.9
99
8
C2
ec~
(5,2
,...
,2)
s~(1
,1,.
..,1
),m
10~
43
.26
11
0.3
98
70
.60
13
0.4
75
10
.52
49
3.4
85
40
.44
73
0.5
52
70
.46
37
0.5
36
3
C2
fc~
(5,2
,...
,2)
s~(1
,1,.
..,1
),s 1
0~
03
.19
29
0.4
00
00
.60
00
0.2
74
80
.72
52
3.3
82
00
.44
56
0.5
54
40
.34
73
0.6
52
7
C2
gc~
(5,2
,...
,2)
s~(1
,1,.
..,1
),s 1
0~
33
.14
07
0.4
00
00
.60
00
0.2
41
50
.75
85
3.3
07
50
.44
44
0.5
55
60
.31
49
0.6
85
1
C2
hc~
(5,2
,...
,2)
s~(1
,1,.
..,1
),c 1
2~
11
.87
84
0.3
99
70
.60
03
0.2
40
50
.75
95
1.9
10
30
.44
64
0.5
53
60
.35
00
0.6
50
0
C2
ic~
(5,2
,...
,2)
s~(1
,1,.
..,1
),c 1
2~
44
.05
67
0.4
00
00
.60
00
0.2
39
70
.76
03
4.5
42
20
.44
18
0.5
58
20
.35
00
0.6
50
0
C2
jc~
(5,2
,...
,2)
s~(1
,1,.
..,1
),m
12~
11
.91
13
0.3
99
90
.60
01
0.2
39
10
.76
09
1.9
36
80
.44
64
0.5
53
60
.35
00
0.6
50
0
C2
kc~
(5,2
,...
,2)
s~(1
,1,.
..,1
),m
12~
43
.88
88
0.4
24
00
.57
60
0.4
87
10
.51
29
4.4
12
60
.44
18
0.5
58
20
.35
00
0.6
50
0
C2
lc~
(5,2
,...
,2)
s~(1
,1,.
..,1
),s 1
2~
03
.32
24
0.4
00
00
.60
00
0.2
41
10
.75
89
3.5
15
80
.44
64
0.5
53
60
.35
00
0.6
50
0
C2
mc~
(5,2
,...
,2)
s~(1
,1,.
..,1
),s 1
2~
32
.98
62
0.4
00
00
.60
00
0.2
39
40
.76
06
3.1
48
20
.44
64
0.5
53
60
.35
00
0.6
50
0
do
i:10
.13
71
/jo
urn
al.p
on
e.0
10
20
75
.t0
05
Optimal Routing
PLOS ONE | www.plosone.org 10 July 2014 | Volume 9 | Issue 7 | e102075
Ta
ble
6.
Eval
uat
ing
the
app
roxi
mat
ion
sfo
rso
me
B-s
ets
.
�~
3a
(1)
ia
(2)
ia
(3)
ia
(4)
i
Se
tH
op
tH
ap
pD
%H
ap
pD
%H
ap
pD
%H
ap
pD
%
B1
a2
.91
65
2.8
21
13
.27
%2
.91
65
0.0
0%
2.8
21
13
.27
%2
.91
65
0.0
0%
B1
b2
.90
04
2.7
97
43
.55
%2
.90
04
0.0
0%
2.7
97
43
.55
%2
.90
04
0.0
0%
B1
c2
.88
50
2.7
76
23
.77
%2
.88
50
0.0
0%
2.7
76
23
.77
%2
.88
50
0.0
0%
B1
d2
.97
33
2.9
73
30
.00
%2
.97
33
0.0
0%
2.9
73
30
.00
%2
.97
33
0.0
0%
B1
e2
.96
72
2.9
67
20
.00
%2
.96
72
0.0
0%
2.9
67
20
.00
%2
.96
72
0.0
0%
B1
f2
.96
09
2.9
60
90
.00
%2
.96
09
0.0
0%
2.9
60
90
.00
%2
.96
09
0.0
0%
B1
g2
.98
78
2.9
83
50
.14
%2
.98
78
0.0
0%
2.9
83
50
.14
%2
.98
78
0.0
0%
B1
h2
.98
47
2.9
79
70
.17
%2
.98
47
0.0
0%
2.9
79
70
.17
%2
.98
47
0.0
0%
B1
i2
.98
16
2.9
75
80
.19
%2
.98
16
0.0
0%
2.9
75
80
.19
%2
.98
16
0.0
0%
Avg
1.2
3%
0.0
0%
1.2
3%
0.0
0%
Min
0.0
0%
0.0
0%
0.0
0%
0.0
0%
Max
3.7
7%
0.0
0%
3.7
7%
0.0
0%
B1
j2
.67
43
2.6
72
10
.08
%2
.67
21
0.0
8%
2.6
72
10
.08
%2
.67
21
0.0
8%
B1
k2
.64
89
2.6
48
50
.02
%2
.64
85
0.0
2%
2.6
48
50
.02
%2
.64
85
0.0
2%
B1
l2
.62
78
2.6
27
80
.00
%2
.62
78
0.0
0%
2.6
27
80
.00
%2
.62
78
0.0
0%
B1
m2
.61
01
2.6
09
80
.01
%2
.60
98
0.0
1%
2.6
09
80
.01
%2
.60
98
0.0
1%
B1
n2
.59
47
2.5
93
80
.03
%2
.59
38
0.0
3%
2.5
93
80
.03
%2
.59
38
0.0
3%
B1
o2
.97
39
2.9
73
20
.02
%2
.97
32
0.0
2%
2.9
73
20
.02
%2
.97
32
0.0
2%
B1
p2
.97
04
2.9
70
20
.01
%2
.97
02
0.0
1%
2.9
70
20
.01
%2
.97
02
0.0
1%
B1
q2
.96
71
2.9
67
10
.00
%2
.96
71
0.0
0%
2.9
67
10
.00
%2
.96
71
0.0
0%
B1
r2
.96
42
2.9
64
00
.01
%2
.96
40
0.0
1%
2.9
64
00
.01
%2
.96
40
0.0
1%
B1
s2
.96
15
2.9
60
90
.02
%2
.96
09
0.0
2%
2.9
60
90
.02
%2
.96
09
0.0
2%
B1
t2
.99
38
2.9
93
60
.01
%2
.99
36
0.0
1%
2.9
93
60
.01
%2
.99
36
0.0
1%
B1
u2
.99
29
2.9
92
80
.00
%2
.99
28
0.0
0%
2.9
92
80
.00
%2
.99
28
0.0
0%
B1
v2
.99
20
2.9
92
00
.00
%2
.99
20
0.0
0%
2.9
92
00
.00
%2
.99
20
0.0
0%
B1
w2
.99
12
2.9
91
20
.00
%2
.99
12
0.0
0%
2.9
91
20
.00
%2
.99
12
0.0
0%
B1
x2
.99
05
2.9
90
30
.01
%2
.99
03
0.0
1%
2.9
90
30
.01
%2
.99
03
0.0
1%
Avg
0.0
1%
0.0
1%
0.0
1%
0.0
1%
Min
0.0
0%
0.0
0%
0.0
0%
0.0
0%
Max
0.0
8%
0.0
8%
0.0
8%
0.0
8%
B2
a2
.99
33
2.9
93
10
.01
%2
.99
31
0.0
1%
2.9
93
10
.01
%2
.99
31
0.0
1%
B2
b2
.99
26
2.9
92
60
.00
%2
.99
26
0.0
0%
2.9
92
60
.00
%2
.99
26
0.0
0%
B2
c2
.99
21
2.9
92
10
.00
%2
.99
21
0.0
0%
2.9
92
10
.00
%2
.99
21
0.0
0%
B2
d2
.99
15
2.9
91
40
.00
%2
.99
14
0.0
0%
2.9
91
40
.00
%2
.99
14
0.0
0%
Optimal Routing
PLOS ONE | www.plosone.org 11 July 2014 | Volume 9 | Issue 7 | e102075
Ta
ble
6.
Co
nt.
�~
3a
(1)
ia
(2)
ia
(3)
ia
(4)
i
Se
tH
op
tH
ap
pD
%H
ap
pD
%H
ap
pD
%H
ap
pD
%
B2
e2
.99
10
2.9
90
90
.00
%2
.99
09
0.0
0%
2.9
90
90
.00
%2
.99
09
0.0
0%
B2
f2
.99
27
2.9
67
40
.85
%2
.99
26
0.0
0%
2.9
67
40
.85
%2
.99
26
0.0
0%
B2
g2
.99
23
2.9
60
21
.07
%2
.99
23
0.0
0%
2.9
60
21
.07
%2
.99
23
0.0
0%
B2
h2
.99
21
2.9
53
21
.30
%2
.99
21
0.0
0%
2.9
53
21
.30
%2
.99
21
0.0
0%
B2
i2
.99
18
2.9
46
71
.51
%2
.99
17
0.0
0%
2.9
46
71
.51
%2
.99
17
0.0
0%
B2
j2
.99
16
2.9
40
51
.71
%2
.99
15
0.0
0%
2.9
40
51
.71
%2
.99
15
0.0
0%
B2
k2
.99
45
2.9
79
40
.50
%2
.97
94
0.5
0%
2.9
94
30
.01
%2
.99
43
0.0
1%
B2
l2
.99
39
2.9
67
90
.87
%2
.96
79
0.8
7%
2.9
93
30
.02
%2
.99
33
0.0
2%
B2
m2
.99
35
2.9
56
31
.24
%2
.95
63
1.2
4%
2.9
92
30
.04
%2
.99
23
0.0
4%
B2
n2
.99
32
2.9
45
01
.61
%2
.94
50
1.6
1%
2.9
91
20
.07
%2
.99
12
0.0
7%
B2
o2
.99
20
2.9
34
31
.93
%2
.93
43
1.9
3%
2.9
89
90
.07
%2
.98
99
0.0
7%
Avg
0.8
4%
0.4
1%
0.4
4%
0.0
2%
Min
0.0
0%
0.0
0%
0.0
0%
0.0
0%
Max
1.9
3%
1.9
3%
1.7
1%
0.0
7%
do
i:10
.13
71
/jo
urn
al.p
on
e.0
10
20
75
.t0
06
Optimal Routing
PLOS ONE | www.plosone.org 12 July 2014 | Volume 9 | Issue 7 | e102075
account the differences (in service rates and squared coefficient of
variation) among the two nodes after the split, which is intuitively
logical as this strategy leads to the highest throughput. Moreover,
the methodology seems to always find the optimal routing vector.Split with Three Branches. Let us now turn to the three-
branch split network, Figure 5. It would be interesting to see to
what extent the optimization algorithm balances the flow over the
three nodes after the split and to what extent this is affected by the
characteristics of the different nodes after the split. Table 3 shows a
selected set of experiments done for this specific case.
Table 3 shows that for the complete symmetric case, that is, set
B2c, Table 3, again the routing probabilities are symmetric, i.e.a i, j~0:333,V (i,j) [ A: For the unbalanced cases in the squared
coefficient of variation (sets B2a, B2b, B2d, and B2e, Table 3), it
can be observed that the routing probability into the two identical
nodes (a 1, 3 and a 1, 4) are close to each other. For the remaining
asymmetrical cases (sets B2f to B2o, Table 3), again the same
conclusion holds. The faster (either in high number of servers or
service rates) or more reliable (in terms of low squared coefficient
of variation) are the nodes, more favored they are, resulting in high
routing probabilities into these nodes.
Complex NetworksThe simple networks discussed so far are interesting as they
make it possible to show the behavior and logic of the optimization
model in the presence of one split. In this section, we will evaluate
some different complex topologies with regard to their routing
probabilities. The first complex network considered is an extension
of the two- and three-branch split networks, as depicted in
Figure 6.
Table 4 gives an overview of a selected set of experiments for
the structure C1. The initial setting is again a balanced case, that
is, c1~5, K1~20, m1~2, ci~2, Ki~2, mi~2, s2i ~1, for
i~2,3, . . . ,7 (set C1a, Table 4). Additional set of experiments
involves unbalancing the service rates mi, the squared coefficients
of variation s2i , and the number of servers ci, for nodes 6 and 7.
With these experiments, we evaluate whether and how the
methodology takes the characteristics of the complete sub-network
after the split into account in determining the optimal routing
vector.
We set up the experiments in such a way that either there are
slow nodes (experiments C1c, C1e, C1g and C1h) or slow
subsystems consisting of three connected nodes (experiments C1b,
C1d, C1f, and C1i). Based on Table 4, we observe that in general
the slower part of the network tends to receive less flow due to a
lower routing probability into the relevant part. When after the
first split in node 1 there is the choice to go to either the fast or
slow subsystem, the faster subsystem is preferred. This is very clear
in experiments C1b, C1d, C1f, and C1i, when the routing
probability always favors the fastest downstream subsystem.
However, if the last nodes are different (experiments C1c, C1e,
C1g, and C1h), the conclusion is different. In all these
experiments, the first split is just exactly half. The imbalance in
the last nodes (i.e. nodes 4, 5, 6, and 7 are different), is completely
absorbed in the routing probability at the immediately preceding
nodes (i.e. nodes 2 and 3). Interestingly, this effect did not
propagate upstream and did not affect the routing at the first split.
Again, we see that the effect of the squared coefficient of variation
on the routing probability is smaller compared to the number of
servers or the service rates.
The second network structure C2 has a more general structure
than the other networks, as seen in Figure 7. Nodes 10 and 12 can
act as a bottleneck node which might become overloaded
depending upon the specific parameters. It is then interesting to
Ta
ble
7.
Eval
uat
ing
the
app
roxi
mat
ion
sfo
rso
me
C-s
ets
.
�~
5a
(1)
ia
(2)
ia
(3)
ia
(4)
i
Se
tH
op
tH
ap
pD
%H
ap
pD
%H
ap
pD
%H
ap
pD
%
C1
a4
.58
10
4.5
81
00
.00
%4
.58
10
0.0
0%
4.5
81
00
.00
%4
.58
10
0.0
0%
C1
b4
.49
78
4.4
71
70
.58
%4
.47
17
0.5
8%
4.4
71
70
.58
%4
.47
17
0.5
8%
C1
c4
.60
02
4.4
74
22
.74
%4
.47
42
2.7
4%
4.5
98
50
.04
%4
.59
85
0.0
4%
C1
d4
.58
08
4.5
80
60
.00
%4
.58
06
0.0
0%
4.5
80
60
.00
%4
.58
06
0.0
0%
C1
e4
.58
26
4.5
80
60
.04
%4
.58
06
0.0
4%
4.5
80
60
.04
%4
.58
06
0.0
4%
C1
f4
.51
93
4.4
53
11
.46
%4
.45
31
1.4
6%
4.4
53
11
.46
%4
.45
31
1.4
6%
C1
g4
.59
64
4.4
53
13
.12
%4
.59
64
0.0
0%
4.4
53
13
.12
%4
.59
64
0.0
0%
C1
h4
.59
75
4.2
37
07
.84
%4
.59
13
0.1
3%
4.5
10
21
.90
%4
.59
73
0.0
0%
C1
i4
.60
03
4.5
24
91
.64
%4
.59
87
0.0
3%
4.5
80
90
.42
%4
.60
00
0.0
1%
Avg
1.9
4%
0.5
6%
0.8
4%
0.2
4%
Min
0.0
0%
0.0
0%
0.0
0%
0.0
0%
Max
7.8
4%
2.7
4%
3.1
2%
1.4
6%
do
i:10
.13
71
/jo
urn
al.p
on
e.0
10
20
75
.t0
07
Optimal Routing
PLOS ONE | www.plosone.org 13 July 2014 | Volume 9 | Issue 7 | e102075
see how the routing probabilities are adapted to avoid or to reduce
the workload in these bottleneck nodes.
Table 5 gives the results for a selection of parameter settings for
network structure C2. The table shows that when node 10
becomes the bottleneck, routing the jobs into the direction of node
10 is avoided by reducing the routing probability at node 3 (always
around 0.4) and node 5 (ranging between 0.0409 to 0.4751). On
the other hand, when the characteristics of node 10 are such that it
is not a bottleneck, then the routing to nodes 5 and 6 is almost 50/
50. Secondly, it is clear that if the last node (node 12) becomes the
bottleneck, only the throughput will be reduced.
Approximations for the Routing ProbabilitiesFrom a managerial point of view, it is interesting to have some
good easy approximations that can be used to quickly set the
routing probabilities. A number of possible approximations for the
routing probabilities in the arc (i,j) [ A, after a split of vertex iinto ni vertexes, can be considered.
a(1)i, j ~
1
ni
,V(i,j) [ A, ð13Þ
a(2)i, j ~
miPVj[d(i)
mj
,V(i,j) [ A, ð14Þ
a(3)i, j ~
ciPVj[d(i)
cj
,V(i,j) [ A, ð15Þ
a(4)i, j ~
cimiPVj[d(i)
cjmj
,V(i,j) [ A, ð16Þ
in which d(i) is the set of succeeding vertexes of vertex i, that is,
d(i):fjj (i,j) [ Ag: Notice that ni is the cardinality of set d(i):The first approximation, Equation (13), is simple but does not
use any information from the ni vertexes after the split. This
approximation only provides an equal spread of the throughput
over the succeeding vertexes. It is expected that this approxima-
tion works well when the nodes after the split are very similar in
terms of service rate, number of servers, and so on. The other
approximations, Equations (14), (15), and (16), do take more
information into account. Equation (16) is believed to be the most
general as it combines information in regard to the speed and the
number of servers. On the other hand, no information about the
squared coefficient of variation is included in none of the
approximations.
Tables 6 and 7 show that the performance of approximation a(4)i
improves as the network becomes more unbalanced (for instance,
cases B1a-B1c are unbalanced, as defined in Table 1, and for these
cases the smallest D% found is for a(4)i , D%~0:00%, on the other
hand cases B1d-B1f are balanced and for them all D% is equal to
0.00%). This approximation of course takes into account the most
information from the nodes after the split (not taking into account
the squared coefficient of variation). If the nodes after the split are
more alike (balanced) then the second approximation becomes
favorable. On the other hand the first approximation a(1)i is
performing acceptable as well and could be preferred due to the
easy implementation.
Conclusions and Final Remarks
In this paper, we examined the optimal routing problem in open
finite acyclic queueing networks with a given general topology and
multiple generally distributed servers. We determined the optimal
routing probability vector that maximizes the throughput of an
arbitrary configured network via a combination of the Generalized
Expansion Method and Powell optimization tool. We presented
numerical results showing the merits of the approach. Approxi-
mations for the routing probability vector are also presented and
evaluated.
We have considered here only the throughput as the main
performance measure. It would also be interesting to evaluate the
behavior of the routing algorithm to minimize the cycle time, the
work-in-process (WIP) or other performance measures. Topics for
future research on the area include the routing in queueing
networks with cycles, e.g., to model many important industrial
systems that have reverse streams of products due to re-work, or
even the extension to GI=G=c=K queueing networks.
Author Contributions
Conceived and designed the experiments: TvW FRBC. Performed the
experiments: TvW FRBC. Analyzed the data: TvW FRBC. Contributed
reagents/materials/analysis tools: TvW FRBC. Contributed to the writing
of the manuscript: TvW FRBC.
Figure 7. Network structure C2.doi:10.1371/journal.pone.0102075.g007
Optimal Routing
PLOS ONE | www.plosone.org 14 July 2014 | Volume 9 | Issue 7 | e102075
References
1. Kock AAA, Etman LFP, Rooda JE (2008) Effective process times for multi-
server flowlines with finite buffers. IIE Transactions 40: 177–186.2. Liu R (2014) Probabilistic decomposition method on the server indices of an
Mj/G/1 vacation queue. Journal of Applied Mathematics 2014: 9 pages.3. Ahmed NU, Ouyang XH (2007) Suboptimal RED feedback control for buffered
TCP flow dynamics in computer network. Mathematical Problems in
Engineering 2007: 17 pages.4. Chen J, Hu C, Ji Z (2010) An improved ARED algorithm for congestion control
of network transmission. Mathematical Problems in Engineering 2010: 14 pages.5. de Bruin AM, van Rossum AC, Visser MC, Koole GM (2007) Modeling the
emergency cardiac in-patient flow: An application of queuing theory. Health
Care Management Science 10: 125–137.6. He X, Hu W (2014) Modeling relief demands in an emergency supply chain
system under large-scale disasters based on a queuing network. The ScientificWorld Journal 2014: 12 pages.
7. Jouini O, Dallery Y, Aksin Z (2009) Queueing models for full-flexible multi-classcall centers with real-time anticipated delays. International Journal of
Production Economics 120: 389–399.
8. Gontijo GM, Atuncar GS, Cruz FRB, Kerbache L (2011) Performanceevaluation and dimensioning of GIX/M/c/N systems through kernel estimation.
Mathematical Problems in Engineering 2011: 20 pages.9. Cruz FRB, Smith JM, Queiroz DC (2005) Service and capacity allocation in M/
G/C/C state dependent queueing networks. Computers & Operations Research
32: 1545–1563.10. Cruz FRB, van Woensel T, Smith JM, Lieckens K (2010) On the system
optimum of traffic assignment in M/G/c/c state-dependent queueing networks.European Journal of Operational Research 201: 183–193.
11. Rahman K, Ghani NA, Kamil AA, Mustafa A, Chowdhury MAK (2013)Modelling pedestrian travel time and the design of facilities: A queuing
approach. PLoS ONE 8: e63503.
12. Khalid R, Nawawi MKM, Kawsar LA, Ghani NA, Kamil AA, et al. (2013) Adiscrete event simulation model for evaluating the performances of an M/G/C/
C state dependent queuing system. PLoS ONE 8: e58402.13. Daskalaki S, Smith JM (2004) Combining routing and buffer allocation problems
in series-parallel queueing networks. Annals of Operations Research 125: 47–68.
14. Garey MR, Johnson DS (1979) Computers and Intractability: A Guide to theTheory of NP-Completeness. New York: W. H. Freeman and Company.
15. Smith JM, Daskalaki S (1988) Buffer space allocation in automated assemblylines. Operations Research 36: 343–358.
16. Dallery Y, Gershwin SB (1992) Manufacturing flow line systems: A review of
models and analytical results. Queueing Systems 12: 3–94.17. Cruz FRB, Duarte AR, van Woensel T (2008) Buffer allocation in general single-
server queueing network. Computers & Operations Research 35: 3581–3598.18. Smith JM, Cruz FRB, van Woensel T (2010) Topological network design of
general, finite, multi-server queueing networks. European Journal of Opera-
tional Research 201: 427–441.19. Smith JM, Cruz FRB, van Woensel T (2010) Optimal server allocation in
general, finite, multi-server queueing networks. Applied Stochastic Models inBusiness & Industry 26: 705–736.
20. Cruz FRB, van Woensel T (2014) Finite queueing modeling and optimization: A
selected review. Journal of Applied Mathematics 2014: 11 pages.21. Gosavi HD, Smith JM (1997) An algorithm for sub-optimal routeing in series-
parallel queueing networks. International Journal of Production Research 35:1413–1430.
22. Kerbache L, Smith JM (2000) Multi-objective routing within large scale facilitiesusing open finite queueing networks. European Journal of Operational Research
121: 105–123.
23. Kerbache L, Smith JM (1987) The generalized expansion method for open finitequeueing networks. European Journal of Operational Research 32: 448–461.
24. Himmelblau DM (1972) Applied Nonlinear Programming. New York: McGraw-Hill Book Company.
25. Smith JM (2008) M/G/c/K performance models in manufacturing and service
systems. Asia-Pacific Journal of Operational Research 25: 531–561.26. Bedell P, Smith JM (2012) Topological arrangements of M/G/C/K, M/G/C/C
queues in transportation and material handling systems. Computers &Operations Research 39: 2800–2819.
27. Powell MJD (1964) An efficient method for finding the minimum of a function ofseveral variables without calculating derivatives. Computer Journal 7: 155–162.
28. Kerbache L, Smith JM (1988) Asymptotic behavior of the expansion method for
open finite queueing networks. Computers & Operations Research 15: 157–169.29. Kimura T (1996) A transform-free approximation for the finite capacity M/G/s
queue. Operations Research 44: 984–988.30. Smith JM (2003) M/G/c/K blocking probability models and system perfor-
mance. Performance Evaluation 52: 237–267.
31. Labetoulle J, Pujolle G (1980) Isolation method in a network of queues. IEEETransactions on Software Engineering SE-6: 373–381.
32. Kleinrock L (1975) Queueing Systems, volume I: Theory. New York, NY, USA:John Wiley & Sons.
Optimal Routing
PLOS ONE | www.plosone.org 15 July 2014 | Volume 9 | Issue 7 | e102075