CC BY
10
CARDINALITY BASED APPROACH FOR RELIABILITY REDUNDANCY OPTIMIZATION OF FLOW NETWORKS
Kumar Pardeep
Department of Instrumentation, Kurukshetra University, Kurukshetra, - 136 119, India Fax: +91 1744 - 238 277, 238 191, 238 035. E-mail: pardeep_kuk@rediffmail.com
ABSTRACT
In flow networks, a reliability model representing telecommunications networks is independent of topological information, but depends on traffic path attributes like delay, reliability and capacity etc.. The performance of such networks from quality of service point of view is the measure of its flow capacity which can satisfy the customers demand. To design a flow network which can meet the desired performance goal, a cardinality based approach for reliability redundancy optimization using composite performance measure integrating reliability and capacity has been proposed. The method utilizes cardinality based criteria to optimize main flow paths and backup paths on priority basis. The algorithm is reasonably efficient due to reduced computation work even for large telecommunication networks.
Keywords: flow networks; capacity; telecommunication networks; heuristics.
1 INTRODUCTION
In networks where nodes and links associate reliability or probabilities of failure are called reliability or probability network models. In such models, reliability optimization of networks is a common topic. Many researchers have emphasized that such evaluations are useful in designing reliable telecommunications networks (Abraham 1979, Theologou & Carlier 1991, Kuo et al. 2007, Schneeweiss 1989). However, this might not be true because constrained reliability optimization of networks has generally been studied with reliability as connectivity measure only. But the practical systems such as computer networks, telecommunication networks, transportation systems, electrical power transmission networks, internet etc. are mostly linked with the performance. The performance of such networks is not only associated with network reliability but also depends on load carrying capacity of each node and link of the network and are termed as flow networks. Therefore, some researchers (Nagamochi & Ibaraki 1992, Varshney et al. 1994, Chan et al. 1997, Soh & Rai 2005) have proposed improved models (termed as capacity related reliability models) to represent performance degradation. These give additional capacities to nodes & links and define the performance for telecommunications networks as the maximum flow determined using standard graph theory. However, it is not true in modern telecommunication networks as the selection of paths to transport flow are decided by routing mechanism and logical links assigned in physical layer. Therefore, all paths of network are not active to carry flow from source to destination. The selection of specific routing paths out of various possibilities is based on certain considerations like reliability, cost and quality. Capacity related reliability (CRR) graph models ignore such actual conditions of telecommunication network design (Hayashi 2008).
Many workers (Wang 2004, Hwang 2005, Ha 2006, Ramirez 2005, 2006) have applied different hierarchical importance criterion such as cutsets and pathsets criticality, Birnbaum importance, component importance, optimal assignment, structural importance and cardinality of pathsets,
cutsets and subsystems etc. for solving reliability redundancy optimization problems of general systems. These criterions are used to devise heuristics for optimal assignments such as more important component gets priority over the less important component for applying redundancy. However, a reliability model of flow networks focuses on traffic path attributes like delay, reliability and capacity etc.. Therefore, CRR model must be modified incorporating attributes of routing paths and logical links assigned in physical layer. In the following sections a novel approach considering the above attributes and also combining the hierarchical importance criterions such as cardinality of pathsets and cutsets, disjoint paths and the cardinality of subsystems for reliability redundancy optimization using composite performance measure (CPM) integrating reliability and capacity has been proposed. The proposed method is capable of addressing the ultrahigh reliability requirements of flow networks efficiently even for large telecommunication networks.
2 COMPOSITE PERFORMANCE MEASURE
A path is a sequence of arcs and nodes connecting a source to a sink. All the arcs and nodes of network have its own attributes like delay, reliability and capacity etc.. From the quality and service management point of view, measurement of the transmission ability of a network to meet the customers demand is very important (Lin 2006). When a given amount of flow is required to be transmitted through a flow network, it is desirable to optimize the network reliability to carry the desired flow. The capacity of each arc (the maximum flow passing the arc per unit time) has two levels, 0 and/or a positive integer value. The system reliability is the probability that the maximum flow through the network between the source and the sink is not less than the demand (Pahuja 2004, Lin 2006, 2007a, b). The presumption that in network any amount of flow can pass through any node or path, is neither valid nor justifiable for real life systems as links and nodes can carry only limited amount of flow. Reliability under flow constraint is a more realistic performance measure for flow networks. A concept of weighted reliability was introduced by Pahuja (2004), which requires that all the successful states qualifying connectivity measure of the network be enumerated and the probability of each success state is evaluated and multiplied by the normalized weight to find out the composite performance of flow networks.
2.1 Notation
al (X) Sensitivity factor of lth minimal path set
bi (x) Subsystem selection factor for ith subsystem with xi components
Cj Total amount of resource j available
gi (x) Amount of resources consumed for jth constraint in subsystem-i with x components
Cji (x) Cost of subsystem i for jth constraint with x components
cg Number of different cardinality groups.
cga(xi) ath cardinality group, a = 1, 2,..., d.
h(.) Function yielding system reliability; dependent on number of subsystems (n) and
configuration of subsystems k Number of constraints, j = 1, 2,..., k
L(x) (L , L ,...,L ), Lower limit of each subsystem i,
m Number of main minimal path sets, l = 1, 2,., m
n Number of subsystems, i = 1, 2,., n
P lth minimal path set of the system
ps (l1,l2, ...,lmmy. priority vector s.t. l1 and lmm are the number of minimal path sets arranged
in decreasing order of path selection parameter al(X).
Qi (x.) Unreliability of subsystem i with xi components.
ri Reliability of a component at subsystem i.
Ri (xi) Reliability of subsystem i with xi components.
Rr Residual resources [total resource available (CJ) - resources consumed (XgJ x.)]
Rs (X) System reliability
S (x) Set of variables that have been used as key-elements in a given decomposed expressions
U (X) (Ux, UX1,..., UXn), Upper limit of each of subsystem i,
x* Optimal solution
x; Number of components in subsystem i; i = 1,2,.. ..n
X A vector (xi,.........xn)
Y Finite set of traffic paths
Z Finite set of cuts of the network
ARi Increment in ith stage reliability when a unit is added in parallel to the ith stage
2.2 Assumptions
Following are the assumptions for the rest of the sections:
1. The system and all its subsystems are coherent.
2. Subsystem structures (other than coherence) are not restricted.
3. The networks are modelled with the help of graphs, the paths (ordered pair of arcs and the members of the ordered pair are reliability and capacity respectively) where in are assigned as the weight of each link.
4. Each link can have only two stages up and down.
5. The network nodes are perfect. If the nodes are not perfect, the method needs to be modified to deal with nodes failures.
6. All component states are mutually and statistically independent.
7. All constraints are separable and additive among components.
8. Each constraint is an increasing function of x. for each subsystem.
9. Redundant components cannot cross subsystem boundaries.
2.3 Composite Performance Measure (CPM)
The weighted reliability measure i.e. composite performance measure (CPM), integrating both capacity and reliability may be stated as by:
CPM ®t,R, (1)
ieS (x)
Where rot is the normalized weight and is defined as:
at, = Capi / Capmax
i.e. the ratio of capacity in the ith state to the maximum capacity (Capmax) of the system and Ri probability of the system being in state Si and is computed as:
R. = p(si}= n Pj * n(2)
J / J k / Slt=0
2.4 Capacity Functions of Networks
The capacity function of different arcs connected in parallel is (Ramirez et al. 2005):
C(X)Par =X Cap, (3)
iex
and the capacity function of different arcs connected in series is:
C(x)Ser = min{ Cap} (4)
The rules for connecting series and parallel arcs to integrate capacity and reliability to give composite performance measure are expressed as:
CR (X)ser = { min Cap,} n r (5)
CR (X )par = X Cap i . U r i=1 i=1
(6)
CPM for series and parallel networks can be defined as:
CPMpar = CR(X)par / Capmax (7)
and
CPMser = CR(X)ser / Capmax (8)
3 PROBLEM FORMULATION AND HEURISTIC METHOD
3.1 Problem Formulation
The general constrained redundancy optimization problem in complex systems can be reduced to the following integer programming problem (Kuo et al. 2001):
Maximise
subject to
and
R(X) = h(R(X),..., Rn(xn)), (9)
^ (10) S gi (Xi ) < Cj, j = 1, 2, k
1 < X. U, i = 1, 2 n.
iex i=1
1
3.2 Proposed Heuristic Method
In real life systems all the arcs are not simultaneously connected to carry flow from source to sink as the selection of paths to transport flow are decided by routing mechanism and logical links assigned in physical layer. Thus in practical systems the entire pathsets are never utilized for transfer of information (Hayashi & Abe 2008). The flow is transmitted through the main path(s) only and in case of failure of main path(s), backup path(s) takes over the task of main path(s). As discussed above the main and backup paths of flow networked are decided by the routing mechanism hence it is presumed that these are known. The proposed algorithm first optimizes the main path(s) and then back up path(s) using cardinality approach for redundancy optimization. The cardinality is defined as number of elements in a mathematical set. On the basis of this definition the cardinality of a subsystem is defined as its frequency of occurrence in all pathsets and cutsets of the network whereas, the cardinality of a pathset is the number of subsystems contained in the pathset. The proposed algorithm first combine the cardinality of different pathsets and cutsets, disjoints paths to form different groups of subsystems to be optimized on priority basis using three phases. Unlike existing heuristics, a switching criterion has been applied to switch from CRR
optimization of one priority group to another priority group. Using this approach network designer can utilize generally limited resources more efficiently (Kumar et al. 2009, 2010a, b, 2011, 2012). The three phases of the proposed method for optimization are:
(i) In the first phase, the path sets having minimum cardinality are given highest priority and the path sets having maximum cardinality are given least priority then all the subsystems having maximum cardinality are found. All highest priority minimal pathsets containing the highest cardinality subsystems form first group. If highest priority minimal path sets are different than that of containing highest cardinality subsystem(s), these path sets along with the path sets containing highest cardinality subsystems are grouped together in the first group. Using this criterion all subsystems of the network are arranged in different groups with decreasing priority importance.
(ii) In the second phase highest selection-factor bi (xt )is computed for the chosen priority group using
b( )_ ar
bi(x) _ k , for each i e cga (x{) (11)
!(gj (x,)/kCj)
j _1
where
AR _ R, (xt) - R, (xt -1) (12)
d is the total number of cardinality groups formed and cga(x) is the ath cardinality group such that a = 1, 2,..., d.
(iii) In the third phase, a redundant parallel subsystem is added to the unsaturated subsystem belonging to the chosen cga (x) with highest selection factor. The three phases are repeated till optimal solution is reached.
After obtaining the optimal solution for the network; calculate the composite performance measure (CPM) for each subsystem of the network. Then evaluate the system reliability using the CPM of the each subsystem. Novelty of the method is that unlike other existing heuristic for complex systems it requires only one selection factor instead two. To determine the total capacity, if system is working normally then capacity for each primary path is ensured otherwise the flow capacity of the primary path is the minimum, and is the summation of the capacities of reserved backup paths that are working. Finally, total capacity is computed by summing the ensured capacities.
3.3 Steps of the Proposed Method
Step1: Find all path sets and cut sets for the network then using cardinality approach:
i) The path sets having minimum cardinality are given highest priority and the path sets having maximum cardinality are given least priority then all the subsystems having maximum cardinality are found.
ii) All highest priority minimal path sets containing the highest cardinality subsystems form first group. If highest priority minimal path sets are different than that of containing highest cardinality subsystem(s), these path sets along with the path sets containing highest cardinality subsystems are grouped together in the first group.
iii) Using this criterion all subsystems of the network are arranged in different d groups with decreasing priority importance.
Step2: Let a = 1; from a = 1, 2,..., d.
Step3: Let xt = 1 for all i; i = 1, 2,., n.
Step4: Compute bi (x. ) using (4.1) for each subsystem belonging to selected cardinality group cga, find i* e cga (xt) such that b**(xt) = max[b (xt)].
Step5: Check, if by adding one redundant subsystem to unsaturated subsystem i*:
i) no constraints are violated and reliability of the subsystem satisfies the stopping criterion and also the capacity of the subsystem is >= flow capacity of path, add one redundant subsystem to unsaturated subsystemi*by replacing x* with x, +1, and go to step 4.
ii) if at least one constraint is exactly satisfied and other are not violated, also and reliability of the subsystem satisfies the stopping criterion and also the capacity of the subsystem is >= flow capacity of path, then add one redundant subsystem to unsaturated subsystem i* by replacing x* with x, +1. The x*= X is the optimal solution. Go to step 6.
iii) if at least one constraint is violated, then remove subsystem i* from further consideration and consider the next subsystem having maximum b*(xt)value and go to step 5.
iv) if all i* e cga (xt )have now been exhausted, check if a < d; then a = a+1 and go to step 4;
v) if a > d then x*= X is the optimal solution, go to step 6.
Step6: Evaluate the composite performance measure (CPM) for each subsystem of the network.
Step7: Evaluate the system reliability using the CPM of the each subsystem.
4 COMPUTATION AND RESULTS
To illustrate the performance of the proposed algorithm a network having six arcs {x1, x2, x3, x4, x5, x6} and five minimal path sets {y1, y2, y3, y4, y5} as shown in the Figure 1 is considered and solved for capacity related redundancy reliability optimization using CPM (7 and 8). System reliability is determined using Bayes method. The network shown in Figure 1 is a bench mark problem, considered by Hayashi & Abe (2008).
Ys / 11/'
X Yi X3 * *
\Xl
Figure 1 Illustration Network
Using Baye's method, the Reliability of the system can be expressed as:
RS(X) = R3 [1- Q6 {I-OQ7Q2XI-Q4Q5)}] + qs[1-(1-r2r5)(1-rIr4)]*q6
(13)
The problem is solved for data given in Table 1. For this first determine all the simple minimal pathsets and cutsets of the Network:
Y = {yi, y2, ys, y4, y5} whereyi = {1, 3, 5}, y2 = {2, 3, 4}, ys = {1, 4}, = {6}, y5 = {2, 5} Z = {zi, Z2, zs, Z4} where zi = {1, 2, 6}, Z2 ={4, 5, 6}, zs ={2, 3, 4, 6}, Z4 ={1, 3, 5, 6}
and then as discussed in Section-3.2 above, on the basis of cardinality of pathsets and cutsets of the subsystems, all the subsystems of the network are arranged in two groups cg1(xi)= {6} having
cardinality 5 and cg2(x)= {1, 2, 3, 4, 5} of cardinality 4. The general problem of constrained reliability redundancy allocation has been solved using the steps discussed in Section 3.3 above. The problem is solved by considering that every flow path has a capacity of 100. The flow network is solved by considering paths y2, ys, and y4 as main paths and y1, y5 as backup paths. The total flow through network at any time should not exceed above 200 in any case. The proposed algorithm gives the optimal solution (2, 2, 2, 2, 1, 3) with system reliability Rs = 0.9578, the optimized subsystem reliability probability Ri and unreliability probabilities Qi are shown in Table 2.
Table 1 Data for Fig. 1
i 1 2 3 4 5 6
ri 0.70 0.75 0.8 0.85 0.70 0.90
Cli 2 3 2 3 1 3
Ci 30
Table 2 Optimized subsystem reliability/unreliability for Fig. 1
i Xl X2 X3 x4 x5 X6
X* 2 2 2 2 1 3
Ri 0.9100 0.9375 0.9600 0.9775 0.7000 0.9990
Qi 0.090 0.0625 0.0400 0.0225 0.3000 0.0009
The capacity of each subsystem of the flow path is taken as 100 and the capacity of flow paths of the network is determined using (3) of proposed approach as:
Cap{yi} = min Cap {1, 3, 5} = min Cap{2*100, 2*100, 100} = 100 Cap{y2} = min Cap {2, 3, 4} = min Cap{2*100, 2*100, 2*100} =200 Cap{y3} = min Cap {1, 4} = min Cap{2*100, 2*100} =200 ^ (14)
Cap{y4} = min Cap {6} = min Cap{3*100} = 300 Cap{ys} = min Cap {2, 5} = min Cap{2*100, 100} = 100
Next the CPM expression (15-20) are derived using (7 and 8) and the value for CPM for an assumed flow of 200 is suppose to pass through the flow path and it comes out to be 1.0000 .
CPM = ^^ [R1R3R5] (15)
Cap max
= (100/200) * 0.91 * .96 * .7 = .3058
= [R2R3R4] (16)
Cap max
= (200/200) * 0.9375 * .96 * .9775 = 0.8797
min CaP ■ r R R ]
—- r R1R 4 ]
Cap max
= (200/200) * 0.91 * .9775 = 0.8895
Cap j
(300/200) * 0.9990
(17)
CPM ,4 = r r 6] (18)
= (1.5) * 0.9990 as 0 < (min Cap i / Cap max ) < 1 so = 1 * 0.9990 = 0.9990
CPM y5 = ^^ [R2R5] (19)
CaP max
= (100/200) * 0.90 * .70 = 0.3281
Composite performance measure integrating the reliability with capacity is calculated as:
CPM Network = 1 - (1 - CPM y1) *(1-CPM y2) * (1 - CPM y3) *(1-CPM y4)*(1-CPM y 5 ) (20)
= 1 - (1 - 0.3058 )(1 - 0.8787 )(1 - 0.8895 )(1 - 0.9990 )(1 - 0.3281 ) s 1.0000
The above result shows that proposed method is capable of optimizing the flow network to transport the desired capacity through the network with highest reliability. However, the selection of main paths and backup paths will affect the quality of composite performance measure. Hence the proper choice of these paths may be done using cardinality criteria (Kumar et al. 2010b) or any other hierarchical measures of importance.
5 CONCLUSIONS
This paper presented a new model for designing reliable flow networks capable of transmitting required flow. The proposed algorithm utilizes the concept of main and backup flow paths. The choice of backup and flow paths is application specific and paths with minimum cardinality may be selected as main path and disjoint paths can be the backup paths. The numerical example demonstrates that the proposed algorithm is fast for designing large, reliable telecommunications networks because the task of optimization is reduced, as only few paths are selected as main paths.
REFERENCES
J. A. Abraham, "An improved algorithm for network reliability," IEEE Trans. Reliability, vol. R-28, pp. 58-61, 1979.
A. Satyanarayana and R. K. Wood, "A linear time algorithm for computing K-terminal reliability in series-parallel networks," SIAM J. Computing, vol. 14, pp. 818-832, 1985.
W. G. Schneeweiss, Boolean FunctionsWith Engineering Applications and Computer Programs. : Springer-Verlag, 1989.
O. R. Theologou and J. G. Carlier, "Factoring & reductions for networks with imperfect vertices," IEEE Trans. Reliability, vol. R-40, pp. 210-217, 1991.
H. Nagamochi and T. Ibaraki, "On Onaga's upper bounds on the mean values of probabilistic maximum flows," IEEE Trans. Reliability, vol. R-41, pp. 225-229, 1992.
P. K. Varshney, A. R. Joshi, and P. L. Chang, "Reliability modelling and performance evaluation of variable link-capacity networks," IEEE Trans. Reliability, vol. R-43, pp. 378-382, 1994.
Y. Chan, E. Yim, and A. Marsh, "Exact & approximate improvement to the throughput of a stochastic network," IEEE Trans. Reliability, vol. R-46, pp. 473-486, 1997.
W. Wang, J. Loman, and P. Vassiliou, "Reliability Importance of components in a complex system", RAM, pp. 6-10, 2004.
Pahuja G. L, "Reliability Evaluation and Optimization Recent and New Approaches ", Ph.D. thesis, Kurukshetra University, Kurukshetra, 2004.
F. K. Hwang, "A Hierarchy of Importance Indices", IEEE Trans. Reliability, vol. 54 (1), pp. 169172, 2005.
S. Soh and S. Rai, "An efficient cutset approach for evaluating communication- network reliability with heterogeneous link-capacities," IEEE Trans. Reliability, vol. R-54, pp. 133-144, 2005.
J. E. Ramirez-Marquez and D. W. Coit, "Composite importance measures for multi-state systems with multi-state components", IEEE Trans. Reliability, vol. 54(3), pp 517-529, 2005
J. E. Ramirez-Marquez, C. M. Rocco, B. A. Gebre, D. W. Coit, and M. Tortorella, "New insights on multi-state component criticality and importance", Reliability Engineering and System Safety, vol. 91, pp. 894-904, 2006.
C. Ha, and W. Kuo, "Multi-Path Heuristic for Redundancy Allocation: The Tree Heuristic", IEEE Trans. Reliability, vol. 55(1), pp.37-43, 2006.
Lin Y. K. 2006, "Reliability of a computer network in case capacity weight varying with arcs, nodes and types of commodity", Reliability Engineering and System Safety, vol. 2(5), pp. 1-7.
S. Y. Kuo, F. M. Yeh, and H. Y. Lin, "Efficient and exact reliability evaluation for networks with imperfect vertices," IEEE Trans. Reliability, vol. R-56, pp. 288-300, 2007.
Lin Y. K. 2007,
a. "Reliability evaluation for an information network with node failure under cost constraint," IEEE Trans. Systems, Man and Cybernetics-Part A: Systems and Humans, vol. 37, no. 2, pp. 180-188.
b. "On a multi-commodity stochastic-flow network with unreliable nodes subject to budget constraint," European Journal of Operational Research, vol. 176, no. 1, pp. 347-360.
Hayashi M. and Abe T. 2008, "Evaluating Reliability of Telecommunications Networks Using Traffic Path Information", IEEE Trans. Reliability, vol. 57(2), pp. 283-294, 2008.
Kumar P., Chaturvedi D. K. and Pahuja G. L, "Heuristic algorithm for constrained redundancy reliability optimization and performance evaluation," Proc. IMechE, Part O: J. Risk and Reliability, vol. 223 (4), 381-386, 2009.
Kumar P., Chaturvedi D. K. and Pahuja G. L,
a. "Heuristic Methods for Solving Redundancy Allocation in Complex Systems", Int. J. of Reliability and Safety, vol. 4, No. 2/3, pp. 285-298, 2010.
b. "An efficient heuristic algorithm for determining optimal redundancy allocation of complex networks", Reliability Theory and Applications, vol. 3(18), pp.15-28, 2010.
Kumar P., Chaturvedi D. K. and Pahuja G. L,"Constrained Reliability Redundancy Optimization of Complex Systems using Genetic Algorithm", MIT Int. J of Electrical and Instrumentation Engineering, vol. 1, No. 1, pp. 41-48, 2011.
Kumar P., Chaturvedi D. K. and Pahuja G. L, "A Heuristic Method for Reliability Redundancy Optimization of Flow Networks", Reliability Theory and Applications, vol. 1, #02(25), pp.6977, 2012.
