The Throughput Flow Constraint Theorem and its Applications

The paper states and proves an important result related to the theory of flow networks with disturbed flows:“the throughput flow constraint in any network is always equal to the throughput flow constraint in its dual network”. After the failure or congestion of several edges in the network, the throughput flow constraint theorem provides the basis of a very efficient algorithm for determining the edge flows which correspond to the optimal throughput flow from sources to destinations which is the throughput flow achieved with the smallest amount of generation shedding from the sources. In the case where a failure of an edge causes a loss of the entire flow through the edge, the throughput flow constraint theorem permits the calculation of the new maximum throughput flow to be done in ) (m O time, where m is the number of edges in the network.In this case, the new maximum throughput flow is calculated by inspecting the network only locally, in the vicinity of the failed edge, without inspecting the rest of the network. The superior average running time of the presented algorithm, makes it particularly suitable for decongesting overloaded transmission links of telecommunication networks, in real time.In the paper, it is also shown that the deliberate choking of flows along overloaded edges, leading to a generation of momentary excess and deficit flow, provides a very efficient mechanism for decongesting overloaded branches. Keywords—networks with disturbed flows; congestion; decongestion; maximum throughput flow; telecommunication networks I. THE NEED FOR A HIGH-SPEED CONTROL OF FLOW NETWORKS Although almost all real networks are networks with disturbed flows, the focus of existing research on flow networks has been exclusively on static flow networks. Research and algorithms related to static flow networks has been presented in [1-3]. The first majorcategory of algorithms for maximising the throughput flow in networks includes the augmentation algorithms which preserve the feasibility of the network flow at all steps, until the maximum throughput flow is attained [4-5].The second major category of algorithms are based on the preflow concept used in [6] and subsequently in [7] and [8]. The central idea behind these algorithms is converting the preflow into a feasible flow. The best of these methods however, have a polynomial running time and do not provide the necessary computational speed for re-optimising the throughput flow in a large and complex network in real time, after an edge failure or congestion. The main reason is that classical algorithms for maximising the throughput flow start from a network with empty edges and do not make use of special properties of the network providing a short cut to determining the maximum throughput flow. The central question for networks with disturbed flows is how to re-optimise the network flows after an edge flow disturbance (caused by edge failure or congestion), so that a new optimal throughput flow is attained quickly.The concept ‘new optimal throughput flow’ means a throughput flow achieved with a minimum reduction of flow production from the flow generators (with a minimum generation shedding). After edge failure or edge congestion, often there exists a possibility for redirecting the flow through alternative paths with non-zero residual capacity, so that a new throughput flow is reached quickly, with a minimum loss of flow. Even for relatively simple networks, it is not obvious how the edge flows should be reset in order to attain the required throughput flow, with a minimum generation shedding. Without an appropriate algorithm, the task of resetting correctly the edge flows in order to attain the new optimum throughput flow is almost impossible, for large and complex flow networks. In addition, the computational time of the algorithm must be within milliseconds, if the algorithm is to be capable of reoptimising the network flows in real time, after a contingency event. For very large networks (>10000 transmission links) an algorithm with approximately linear average running time in the size of the network is needed. The lack of optimisation of the network flow after a contingency event leads to a severe disruption of the flow, suboptimal performance and loss of throughput flow. The importance of dynamically rerouting the traffic in telecommunication networks has been stressed in [9]. Despite the critical importance of the problem related to re-optimising the flows upon disappearance of an edge due to failure, it is difficult to find a relevant theoretical discussion.Such a discussion would provide the necessary short cut speeding the performance of algorithms calculating the re-optimised throughput flow. This problem has been mentioned in question 6.35b from [1], where the authors propose to the reader to show that after an overestimation of the capacity of an edge by k units, the labelling algorithm can re-optimise the maximum flow in O(km) time, where m is the number of edges in the network. (IJACSA) International Journal of Advanced Computer Science and Applications, Vol. 5, No. 3, 2014 12 | P a g e www.ijacsa.thesai.org Disregarding the fact that the running time O(km) of the labelling algorithm is too slow for large k, the direct application of the labelling algorithm to solve this problem leads to sub-optimal solutions. This can be demonstrated immediately with the network in Fig.1, where the first number on the edges denotes the throughput capacity of the edge and the second number denotes the actual flow through the edge. Initially, the throughput flow has been maximised by using the labelling algorithm. The path (1, 2, 7) has been augmented with 40 units and path (1, 3, 6, 7) with 60 units. There are no more augmentable s-t paths and according to the FordFulkerson theorem [4] the throughput flow in the network is the maximum throughput flow. Now suppose that the capacity of edge (3,6) has actually been overestimated by 30 units and the actual capacity of the edge is only 30 units. In this case, the labelling algorithm reoptimises the flow by diminishing the flow along the s-t path (1,3,6,7) by 30 units.However, the total throughput flow in the network from the source s to the sink t, obtained by the labelling algorithm, is 70 units, which is a sub-optimal value (Fig.1b). If before diminishing the flow along the path (1,3,6,7) an additional operation was performed, the total throughput flow would be 80 units, not 70 units. Due to constraining the flow along edge (3,6) by 30 units a “momentary excess flow” appears at node 3 and a “momentary deficit flow“ appears at node 6. The network should be augmented first with the momentary excess flow of 30 units at node 3 aimed to cancel first the momentary deficit flow of 30 units at node 6. As can be seen, the maximum of 10 units momentary excess flow can be sent from node 3 to node 6 through path (3,4,2,5,6). After cancelling 10 units of flow, the remaining momentary excess flow at node 3 is 20 units and the remaining momentary deficit flow at node 6 is also 20 units. Now, by using the labelling algorithm, the momentary excess flow and deficit flow can be reduced to zero by diminishing the flow along the s-t path (1,3,6,7) by 20 units. The result is the network in Fig.1c where the total throughput flow is 80 units, which is the maximum possible throughput flow. A similar deficiency is present in the algorithm reported in [10], treating the problem of maximising the flow in a static network by starting from a network where all edges are fully saturated with flow. This causes unbalanced excess and deficit nodes to appear in the network. The sum of the flows going into an excess node is greater than the sum of the outgoing flows while for a deficit node, the sum of the ingoing flows is smaller than the sum of the outgoing flows. The essence of the draining algorithm presented in [10]is to cancel excess flow with a deficit flow by augmenting paths starting from excess nodes and ending at deficit node. The process of cancellation of excess and deficit flow in [10] was done only in a network with a back circulation edge, connecting the sink with the source. In [11], it was shown that this approach leads to suboptimal solutions where the obtained throughput flow is feasible but not maximal. It has also been demonstrated in [11] that to achieve an optimal solution, there is a need of two distinctive stages. In the first stage, cancelling of excess and deficit flow is done in a network without a circulation edge. In the second stage, draining of excess and deficit flow is done in a network with a circulation edge. In short, applying the labelling algorithm without an intermediate stage consisting of cancelling as much as possible excess and deficit flow, results in sub-optimal solutions. Fig. 1. Applying the labelling algorithm for re-optimising the flow after an overestimation of the capacity of an edge results in a sub-optimal solution Component failures in flow networks and congestion are inevitable. These events lead to disappearance of flow capacity and the expected magnitude of the throughput flow from sources to destinations may not be guaranteed. As a result, the quality of service received from the network (which is a key performance characteristic) can be affected seriously. These problems are particularly acute for telecommunication networks, oriented towards media applications, for transportation networks and power distribution networks, because they all require a high throughput flow rate. Selecting the shortest path for a data transfer, as it is commonly done [12] is often far from optimal. It is a common-sense strategy which often leads to overloading and congestion of network sections, and ultimately, to a low throughput flow. Consequently, the objectives of this paper are: (i) to present a theoretical analysis of the important problem related to the flow constraint arising in the case of edge failures or congestion;(ii) to use the analysis for improving the efficiency of calculation of the new optimal throughput flow after failures or congestion of edges in the network and (iii) to achieve the new throughput flow in the network with a minimum generation shedding. II. AN NEW THROUGHPUT FLOW WITH MINIMUM GENERATION SHEDDINGAFTER FAILURE OR CONGESTION OF SEVERAL EDGES A flow network can always be modelled by a directed graph G = (V,E) consisting of a set of nodes V and a set of edges. The network flow is said to be feasible, if the next two conditions are fulfilled. At each node i v , different from a source or a sink, the flow conservation law holds (equation 1).          m i k i m v f v k f ) , ( ) , ( (1) (IJACSA) International Journal of Advanced Computer Science and Applications, Vol. 5, No. 3, 2014 13 | P a g e www.ijacsa.thesai.org The flow conservation law simply states that the sum of the edge flows    k i v k f ) , ( entering node i v is equal to the sum of the edge flows    m i m v f ) , ( leaving the node. The second condition involves the capacity constraints imposed by the edges: ) , ( ) , ( j i c j i f  (2) The capacity constraint condition simply states that the flow ) , ( j i f through edge (i,j)cannot exceed the capacity ) , ( j i c of the edge – the maximum flow that the edge permits. The performance of an algorithm for re-optimising the network flows after failure or congestion of edges can be increased significantly in comparison with classical algorithms, which always start the maximisation of the throughput flow from a network with empty edges. This can be done by using the important circumstance that after a disturbance of the flow along a particular edge, the rest of the edges are not empty but are already fully or partially saturated with flow. An algorithm which starts the reoptimisation from a network with edges which are fully or partially saturated with flow avoids the augmentation of all feasible paths and has a clear advantage to an algorithm which starts the optimisation from a network with empty edges. In the case of a single edge failure, a method for reoptimising the flows has already been outlined in [13]. However, the critical question related to eliminating the overloading and congestion along branches of a flow network, with minimum generation shedding,was not discussed in Ref.[13]. Finally, Ref.[13] treats only the special case where the throughput flow in the network is the maximum possible throughput flow. In communication networks, electrical networks and transportation networks however, the throughput flow is rarely the maximum possible throughput flow. For real networks the central issue is to re-optimise the network flow in such a way that the contingency event causes a minimum flow generation shedding. In this sense, the notion ‘optimum throughput flow’ used here stands for the restored new feasible throughput flow in the network attained with the smallest decrease of flow generation (generation shedding). As we shall see later, the deliberate choking of flows along overloaded edges, leads to a generation of momentary excess and deficit flow and provides a very efficient mechanism of relieving overloaded branches of the network. In this respect, it is important to state and prove a result related to the magnitude of the optimal throughput flow after the flows along several edges have been constrained (choked) to a particular level. After the choking of the flow along an edge (e.g. edge ( i e , i d )), from the initial level ) , ( i i d e f to the level ) , ( ' i i d e f ( ) , ( ) , ( ' 0 i i i i d e f d e f   ), the network flow is disturbed at nodes i e and i d to which the edge ( i e , i d ) has been connected. The flow along edge ( i e , i d ) may be fully choked because of edge failure ( 0 ) , ( '  i i d e f ) or partially choked ( ) , ( ) , ( ' 0 i i i i d e f d e f   . If edge ( i e , i d ) is not empty, after the choking of its flow, a momentary excess flow appears at one of the nodes (node i e ) equal to the amount of choked flow ) , ( ' ) , ( i i i i d e f d e f  along the edge. In other words, the sum of the edge flows going into node i e is greater than the sum of the edge flows leaving the node. This difference will be referred to as ‘momentary excess flow’ i mef :            m i k i i m e f e k f mef 0 ) , ( ) , ( (3) and node i e will be referred to as momentaryexcess node. Alternatively, after choking the flow along edge ( i e , i d ), momentary deficit flow will be created at node i d , equal to the amount of choked flow ) , ( ' ) , ( i i i i d e f d e f  along the edge( i e , i d ). The sum of the edge flows going into node i d is smaller than the sum of the edge flows leaving node i d . The difference between the sum of the ingoing flows and the sum of the outgoing flows is negative, and will be referred to as momentary deficit flow i mdf :            m i k i i m d f d k f mdf 0 ) , ( ) , ( (4) Accordingly, node i d will be referred to as momentary deficit node. After choking the flow along n edges, 1 M momentary excess nodes i e , with momentary excess flows i mef (i=1,..., 1 M ) and 2 M momentary deficit nodes j d with momentary deficit flows j mdf (j=1,..., 2 M ), will be created. In general, 2 1 M M  because a momentary excess flow at a particular node from choking the flow along a particular edge may have been compensated or turned into a deficit flow by the momentary deficit flow from choking the flow along another edge incident to the node. The quantity 0 1