Different criteria of dynamic routing

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Procedia Coputer Science Volue 66, 2015, Pages 166 173 YSC 2015.

Acadey of Sciences kurochkin@iitp.ru, greenjak@iitp.ru Abstract The atheatical odel of functioning of a telecounication network is developed.

1 Procedia Coputer Science Volue 66, 2015, Pages YSC th International Young Scientists Conference on Coputational Science Different criteria of dynaic routing Kurochkin 1*, Grinberg 1 1 Kharkevich Institute for Inforation Transission Probles of ussian Acadey of Sciences kurochkin@iitp.ru, greenjak@iitp.ru Abstract The atheatical odel of functioning of a telecounication network is developed. The analysis technique of the telecounication networks, loading of networks revealing direct dependence on routing strategy is ipleented. The priary goals which can be solved: check of efficiency of strategy of routing; deterination of vulnerabilities in a telecounication network; execution of estiation and the coparative analysis of various strategies of routing. esults of experients on coparison of efficiency of heuristic algoriths of the routing directed on the axiu increase in the general traffic on networks are presented. Coparing of results of atheatical siulation by several criteria was allowed to range heuristic algoriths on a level of fitness to dynaic filling of telecounication networks. Keywords: Dynaic routing, telecounication network, rational use of network resources, ulticoodity proble, inial cost path algorith 1 Introduction In the paper we present our results of studies of the proble which is very iportant in telecounication industry. We consider telecounication network consisting of coutation nodes and counication links of liited capacities. Then, suppose that a nuber of network subscribers pairs generate a sequence of essages (requests) which have to be sent fro one subscriber to another. How should these essages be routed to axiize the nuber of fulfilled requests? Prior to atheatical foralization we will present brief survey of ethods of siilar probles' solving. We think that well-known in the network flow theory so called ulticoodity proble (Fulkerson & Ford, 1962), (Hu, 1970). It is stated as the adissibility proble for coodity specified intensity flows organization in a network with specified capacitated links. It is true that in such traditional stateent the requireents for intensity of coodity flows aong subscribers are * asterinded EasyChair and created the first stable version of this docuent Created the first draft of this docuent 166 Selection and peer-review under responsibility of the Scientific Prograe Coittee of YSC 2015 c The Authors. Published by Elsevier B.V. doi: /j.procs

2 Different criteria of dynaic routing... specified at once, iediately. Our proble will be reduced to a traditional stateent, if all requests aong subscribers are suarized at a certain oent and the flow distribution proble is solved in line with the adissibility proble. It should be noted that there are no special effective ethods of ulticoodity proble solving, in respect of coputation it is a linear prograing proble. The queuing network ethod is the ain ethod of telecounication network analysis. any works are devoted to this ethod (Ivnitsky, 2004) (artynov, Vatutin, & Titiov, 2014). Nevertheless we don t know works within the fraework of this ethod devoted to search of routing algoriths axiizing or at least increasing network total capacity as copared to any known algorith. The prevailing routing algorith in Internet is the iniu path-length algorith (used for exaple in the network layer IP-protocol) or the iniu cost path algorith where costs are assigned to counication lines in accordance with various criteria delay factor, transission bandwidth, reliability etc. The nuerical values of criteria in the OSPF network layer protocol are called etrics. Qualitative ethod presented in this work includes the following. At each certain oent a telecounication network is characterized by links residual capacities. Next inforation channel should be developed so that as uch as possible of resource could be left in the network (Grinberg, Kurochkin, & Korkh, TPSA, 2012). By resource we shall eat two criteria link residual capacity and inial-cut inial capacity of all subscriber pairs. Accordingly two groups of algoriths will be offered. 2 Proble stateent and brief description As a atheatical odel of telecounication network we take a connected un-oriented graph, where vertices represents network coutation nodes and edges (Grinberg & Kurochkin, The study of the results of atheatical odeling of networks with sequential filling of the cluster topology, 2009) (Grinberg & Kurochkin, atheatical odeling of dynaic networks of sequential filling of streas due, 2009). Initially, i.e. the epty network is characterized by edge capacities. Network subscribers are represented by a subset of vertices (further we'll call the poles). Those network nodes that are not included in the poles set are coutation ones. On the subscribers request a path (counication channel) is developed between a pole pair passing along the graph edges and a coodity unit-intensity flow is allowed along this path. Hereupon capacities of the edges, along which the path passes, are reduced by a unit. Then another pair of poles is selected and the procedure is repeated. The proble is: how to select paths between pole pairs to fulfill axiu nuber of requests. In other words which should be the path-tracing algorith - routing algorith to fulfill axiu possible nuber of requests for all types of networks and all variants of requests aong poles. Let us call algoriths, realizing path selection in sequence as per each request, sequential algoriths. It is supposed that it is ipossible to change the selected path. Let us call ulticoodity proble solving algoriths, when edge flows and paths identification is ade in accordance with pole interchange intensity specified suarized requireents, synchronous algoriths. One of sequential algoriths is the iniu path-length algorith (as per edge nuber). Let us call this algorith a siple algorith. One of synchronous algoriths is the linear prograing algorith or LPalgorith. The LP- algorith is absolute one in the sense that in the network load of requireents can be coplied with when and only when adissibility proble is solved. Let us call the sequential algorith optial one, if ipossibility to fill the next request per this algorith eans adissibility proble insolubility for the entire load of requireents. We think that it is hardly possible to develop an optial sequential algorith due to the fact that at each certain oent there is uncertainty of requireent final distribution aong pole pairs. The 167

3 Different criteria of dynaic routing... objective of the present work is to offer sequential algoriths increasing network suarized traffic as copared to a siple algorith. Two groups of algoriths will be offered. One of these is based on the concept that the next path to be developed along the edges with this oent axiu residual capacities. Another group is based on coputation of capacities of inial cuts aong all pole pairs on the basis of data about network edge residual capacities as well as on acceptance of probability law for requireent distribution aong pole pairs. 3 Sequential filling algoriths Let G be a network consisting of N vertices A 1, A 2,,A N, where A i, i=1,2, N 1, N 1 N are poles, and K edges B 1,,B K, with b k, k=1,2,,k, capacities. Let us arrange soehow all pole pairs. The total nuber of various pole pairs in the G network is equal to = N 1 (N 1-1)/2. Let us assue that coodity flow aong pole pairs organization requireents follow the probability law, i.e. there are nonegative nubers p, = 1,2,,, such that p 1. These nubers are probabilities of request for coodity flow between the -th pole pair in coon requests flow. We suppose also that request copletion eans organization of unit intensity flow. 3.1 Fundaental inial-cut algorith Let us introduce the following designations. capacity of the inial cut between the -th pole pair equal, in accordance with the axial flow and inial cut, to the axial flow between this pole pair; inial cut capacity average aong all pole pairs. Let us generate the following value, which will be later called criterion or irregularity easure : 1 2 ( p), 1 1 (1) This value is the isatch easure between inial cuts (noralized in a certain anner) and probabilities of this pole pair requests. The ore isatch is the bigger value is. eanwhile it is clear that one should expect that the better atched inial cuts and probabilities are, the ore requests can be filled. Thus the next path is to be selected so that this value should be inial in the network with edge capacity residual values. This is equivalent to this value increent upon the next path selection should be inial (in the algebraic sense). In the special case of request equiprobable distribution as per pole pairs, i.e. with p = 1/, (1) it is siplified and takes on the following for: 2 ( 1). (2) 1 (The 1/ 2 constant factor is not essential and is not taken into consideration.) The next request fill is identification of the path between the pole pair and allowing the unit intensity flow along it. Herewith the G network is changed as b k. arc capacities are changed. This change is to result in the χ value inial increent. Let us express the χ value and its increent in ters of b k. value. For this purpose we will need to know all inial cuts aong all pole pairs in the G network. 168

4 Different criteria of dynaic routing... Let us assue that b, s 1,2,..., S, the s-th edge of the l-the inial cut between the -th l s pole pair, S l this cut edge nuber. Then l L Sl 1 b, (3) L l 1 s 1 where L total nuber of inial cuts between the -th pole pairs. Let us convert this forula by defining u, 1,, k 1, K nubers as such, which designate the nuber of various k inial cuts between the -th pole pair, which include the B k. arc. Then it is evident that k k L k 1 l s K 1 u b. (4) With account of (4) the network irregularity (1), (2) appears to be expressed in ters of the b k. values. Upon the next eleentary requireent is filled the G network is changed as related to the b k, value change, therefore the value will also be changed. In the case if the b k value relative change is inor, the network irregularity change is deterined by the next forula: Later we have: where u ( ) b, K K b k k k1bk k1 bk 1 b (5) k , 1 (6) 1 uk rk, (7) b L k 2 1 ( p), k rk, rk rk. L 1 K bk rk rkp r 2 k rk k1 1 1 (9) Equation (9) iplies that network irregularity easure increent is given as the linear for as per arc capacity increents. The proble is to ake this increent inial (in the algebraic sense, i.e. increent negative values are possible) while filling the next eleentary requireent. Thus we coe to the proble of inial cost path search. If the eleentary requireent flow is equiprobable for all pole pairs, then (9) is siplified and becoes:, (8) 169

5 Different criteria of dynaic routing... K bk r 2 k rk k1 1. (10) Let us write out explicitly the forula with account that for any path all bk increents are negative (or equal to zero): 2 1 Wk rk rkp r, 2 k rk 1 (11) 1 in the general case and 2 1 Wk r. 2 k rk (12) 1 for requests equiprobable distribution. The forulas (11), (12) include values, fully deterined by network edge capacities, which are changed upon each next eleentary requireent fill. Thus in general ters edge cost total recalculation is needed at each step for deterination of the optial path according to this criterion. [4][5] 3.2 Fundaental edge algorith We want to recall that this algorith idea is that each next path should be allowed along axiu capacity network edges. The proble of this section is to foralize this idea so that this path search appeared to be inial cost path search. Let us introduce the following definitions. Let us distribute all network edges by classes, in each of which let us place the sae capacity edges, and let us arrange all classes in the order of included edge capacity increase. Let us assue that B 1,, B q,, B Q are designations of introduced classes (the fact the edge class designation coincides with the G network edges theselves designation as a atter of fact there s no risk of abiguity), K 1,, K Q are edge nuber in these classes, b 1,, b Q are the capacities value of the edge included in these classes. Now let C be the path fro one pole to another and this path includes c 1 edges of the B 1, class, c q N c Q edges of the B Q class. By definition all c q, c q 0, are integers, oreover, 1, as any q1 path cannot include ore than с edges. Let us treat each path as a Q-bit nuber, i.e. c 1, c 2,, c Q, where each digit is an integer less or equal to N-1. It enables us to arrange eleents of path set in nuerical order. Each path corresponds to unique nuber, but the sae nuber can represent different paths. Now it is possible to state path optiality criterion with the next request fill the inial path is the optial path, i.e. the path deterining the inial nuber The proble is that it is necessary to assign such costs to the edges that the nuber-path having a unit in a bit and zero in previous ones, should be ore than any other path-nuber having zero in all these bits. One of the possible ethods of such costs deterination is the following one: Qq Wq ( N 1), q 1,... Q The path cost in this case is just a nuber, which it deterines, understood to be a N-1-based positional nuber. Q (13) 170

6 Different criteria of dynaic routing Other inial-cut algoriths Preliinary calculations in accordance with the (12) forulas showed that edge costs can be both positive and negative, herewith negative cycles are not rare ones. In these conditions inial cost path selection is the NP-full proble which is inacceptable for nuerical experients. Therefore we introduce fundaental inial-cut algorith odification, which is as follows. Aong all edges having in accordance with (12) forulas negative values we will select the axiu per odulus value and will designate it with W 0. Let us introduce new costs in accordance with the forula W W W0, 0 k k (14) Such transforation akes all costs positive and deletes the probles of inial cost paths search, but it is of course arbitrary. Such transforation effectiveness can be justified or invalidated only by nuerical experients. Let us call the algorith based on such transforation the additive incut algorith. Another algorith is based solely on the siple concept that if a G network edge enters the inial cuts, then its cost should be as higher as less is the inial capacity of all inial cuts, where this edge enters. Soe preliinary researches resulted in the following ethod of edge cost deterination: 4 W N 1 ( h 1), (15) where h is the serial nuber of the, inial cut value, which are ordered in accordance with capacity increase and is the total nuber of various per capacity inial cuts. Let us call this algorith epirical incut algorith. 3.4 Other edge algorith The fundaental edge algorith is the contrasting algorith, in which the next path is allowed along less capacity edges only in the case, if it is ipossible to allow it along ore capacity edges. This algorith can be softened by introducing, like in epirical inial-cut algorith, continuous function of weight assigning. I.e. let us assue that edge costs are deterined by the forulas 4 W N 1 ( T a 1), (16) where a is the serial nuber of the class of the edges, which are arranged in accordance with capacity increase and T is the nuber of edge classes. This function was selected in accordance with the principle: the less is the arc capacity, ore expensive it is. This algorith was called the epirical edge algorith. 4 esults and discussion We developed a coputer odel of all ain stages of telecounication network functioning generation of nonoriented graph with specified nuber of vertices, poles, edges and capacities thereof, generation of request flow, coputation of the network irregularity criteria, edge cost calculation for each algorith there is its own calculation ethod deterination of inial cost path, unit intensity flow passing and corresponding change of the network edge capacities. The nuerical experient in general for one network is the nuerical realization of this network filling process independently for each algorith. The general factor for these experients is, first, the initial ( epty ) network and, second, the requests sequence. In all presented in this work experients the request sequence was equiprobable for all pole pairs. The experient to be done in two odes static and dynaic. In the static ode the path developed between the pole pair and the unit flow along it are kept until the experient copletion. 171

7 Different criteria of dynaic routing... Thus such network resource like all its edges capacity onotonically decreases. In the dynaic ode the counication channel lifetie is finite and follows the probability law we accepted the Gaussian offset law. In this ode the network filling process can include (or even copletely consist of) stationary areas with failure various levels. In the stationary ode filled requests nuber to first failure is registered as well as total nuber of filled requests and total nuber of failures, and the experient is copleted either when failure intensity is over a certain critical level or upon tie expiry, i.e. of total nuber of the network requireents. Networks with topology connected clusters are considered and stochastic topology of the Network consist of nodes, have the initial capacity of arches fro 30 to 500 and have 7-45 couples poles (couples a source drain). Filling is ade by single flows. atheatical odeling of process of filling of a network happens before the following criteria: criterion of the first refusal (carrying out the next deand is ipossible, one of couples a source drain is incoherent; iniu fro a set of the iniu cuts between couples of poles it is equal 0); criterion of full refusal (drains aren't achievable fro sources, for all couples of poles; all iniu cuts between couples of poles are equal 0); an exit to the stationary ode when the criterion of full refusal isn't achievable, but rather large nuber of deands is already processed Dynaic filling of a telecounication network assues that deands through certain tie release network resources, that is have soe tie of life after which the capacities of arches belonging to the carried-out flow of the deand increase by the size of the carried-out strea. Depending on distribution of tie of life of deands filling of a network can proceed in several odes, however the stationary ode when siultaneous nuber of deands in a network big, soeties there are refusals is ost of all interesting, the size of the axiu strea is soeties reached, but ass refusals aren't present; unevenness easure size depending on the carried-out deands a constant. By results of filling of networks with various algoriths of filling it is possible to draw a conclusion that use of algoriths of incut and arc approach will be ore effective for networks with a big initial capacity of arches (ore than 150) and for a sufficient large nuber of couples of poles (ore than 15). Duration of the stationary ode and tie of effective network functioning is ore when using incut algoriths. 5 Conclusions The ethodology of the analysis of telecounication networks (and not only, and in general, strea networks) allowing to investigate characteristics of a network in general, in particular, its total capacity depending on application of this or that policy of routing is created. The ethodology is realized in the for of the atheatical odel iitating process of consecutive filling of a telecounication network with counication channels by eans of various algoriths of routing. Algoriths of routing are original and in fact heuristic. All of the are directed on the axiu increase in a total general strea on a network and divided into two groups inial cut in which belonging of edges of a network to the iniu cuts between the corresponding couples of poles, and arc in which only the current capacity of edges of a network is considered is considered. eferences Fulkerson, D.., & Ford, L.. (1962). Flows in networks. New Jersey: Princeton University Pres. 172

8 Different criteria of dynaic routing... Grinberg, Y.., & Kurochkin, I. I. (2009). atheatical odeling of dynaic networks of sequential filling of streas due. Probles of coputations in the distributed edia: Proceedings of ISA AS(vol.46), p Grinberg, Y.., & Kurochkin, I. I. (2009). The study of the results of atheatical odeling of networks with sequential filling of the cluster topology. Probles of coputations in the distributed edia: Proceedings of ISA AS(vol.46), p Grinberg, Y.., Kurochkin, I. I., & Korkh, A. V. (2012). Theory and Practice of the Syste Analysis. Selection of clusters in networks with dynaic flows filling, vol.1, pp ybinsk. Hu, T. C. (1970). Integer Prograing and Network Flows. Ivnitsky, V. A. (2004). Theory of queueing network. oscow: Physatlit. artynov, I. A., Vatutin, E. I., & Titiov, V. S. (2014). Eighth World Conference on Intelligent Systes for Industrial Autoation. Hardware oriented classification of binary relations of graph-schees of parallel algoriths. Tashkent. 173

Utility-Based Resource Allocation for Mixed Traffic in Wireless Networks

IEEE IFOCO 2 International Workshop on Future edia etworks and IP-based TV Utility-Based Resource Allocation for ixed Traffic in Wireless etworks Li Chen, Bin Wang, Xiaohang Chen, Xin Zhang, and Dacheng