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International Journal of Computer Information Systems, Vol. 5, No.2, 2012 An Integrated Approach To Optimizing Performance In Transaction Network Serving Mixed Flows 1 2 A.RARAJU K.SRINIVAS 1 2 M.Tech (CSE) Kottam College of Engineering, Professor and HOD, Kurnool. raraju22@gmail.com Kottam College of Engineering, Kurnool, kipgs2008@gmail.com Abstract—In this work, we study the control of structure of the solution, and forces us to develop communication networks in the presence of both a joint congestion control and load balancing inelastic and elastic traffic flows. The mechanism that is fully distributed, and achieves characteristics of these two types of traffic differ high throughput and good delay characteristics. significantly. Hence, earlier approaches that focus on homogeneous scenarios with a single traffic type Our goal is to balance the load of the inelastic are not directly applicable. We formulate a new traffic in the network such that the elastic traffic network optimization problem that incorporates intelligently exploits the time varying residual the performance requirements of inelastic and capacity (the link capacity minus the capacity elastic traffic flows. The solution of this problem needed to serve the inelastic flows) at each link in provides us with a new queueing architecture, and the network. To see the potential gains of such an distributed load balancing and congestion control interaction, consider the network shown in Figure algorithm with provably optimal performance. In 1, which serves one inelastic and one elastic flow particular, we show that our algorithm achieves the over links of capacity 20. Assume that the dual goal of maximizing the aggregate utility gained by the elastic flows while satisfying the inelastic flow has a fixed rate of 20 and has two demands of inelastic flows. Our base optimal routes to divide its traffic over as shown in the algorithm is extended to provide better delay figure. It can be seen that the rate distribution performance for both types of traffic with minimal decision of the inelastic flow will significantly degradation in throughput. It is also extended to affect the elastic flow performance. If it divides the practically relevant case of dynamic arrivals its traffic equally amongst the two routes as in and departures. Our solution allows for a Figure 1, the elastic flow cannot achieve a rate controlled interaction between the performance of more than 10. However, if the inelastic flow can inelastic and elastic traffic flows. This performance steer more of its traffic over the less congested can be tuned to achieve the appropriate design tradeoff. The network performance is studied both route, more resources become available to the theoretically and through extensive simulations. elastic traffic and it can achieve rates close to 20 as shown in Figure 2. With this intuition, we want I. INTRODUCTION to design a dynamic algorithm that automatically Over the last several years, we have witnessed the adapts the operation of inelastic and elastic flows development of increasingly sophisticated to get the optimal performance. This requires a optimization and control techniques to address a solution that seamlessly and distributively variety of resource allocation problems for balances the load of the inelastic traffic across the communication networks (e.g. [2], [11], [19], [1], network as well as injects enough elastic traffic [10], [15], [23], [6], [21], [26], [13], [7], see [17], into the network so that no capacity is wasted [8] for an overview). Much of this investigation while preventing network overloading. has focused primarily on controllable or elastic Fig. traffic. Thus, it is imperative that one develop efficient resource allocation strategies to jointly manage both inelastic and elastic traffic. Integration of elastic and inelastic flows in single- hop wireless systems has been studied [24], [3], [22], and in [9] it has been extended to a multiple-hop network, however with the Fig:1.Fixed inelastic Traffic restriction of every flow having a single route. Fig:2.Controllable inelastic Traffic The availability of multiple routes, which is studied in this paper, significantly changes the August Issue Page 48 of 68 ISSN 2229 5208 International Journal of Computer Information Systems, Vol. 5, No.2, 2012 We begin first by providing the system model [t])rЄRi ,we note that Ai[t] denotes the number of and general assumptions. We then formulate a packets generated by flow fi while (xi(r)[t])rЄRi problem that attempts to maximize the utility of describes the number of packets injected into the the elastic flows in the network subject to the network to traverse each of the available routes of constraint that the data requirements of the flow fi. Thus, Ai[t] is an uncontrollable stochastic inelastic flows are met. We solve this problem via process describing exogenous arrivals, whereas a two-step approach. First, we solve a simple (xi(r)[t])rЄRi is controllable by the network version of the problem when the inelastic flow algorithm. For notational convenience, we define rates are deterministic. We then use the insights gained from that framework and extend the solution to the more general stochastic case. We then extend the work in two practically important directions. The first is to develop a virtual queue to denote the total number of inelastic packets on based solution that allows us to achieve low link l for a given xi[t]. Elastic Flow: We let fe delays with a nominal and controllable sacrifice denote an elastic flow in the network with source in the throughput of the elastic flows. The second se and destination de. We assume that each is to extend the solution in the presence of flow elastic flow fe is associated with a single route arrivals and departures, where certain elastic Re; and we let xe[t] be the the number of injected flows may be very short and may leave the packets of flow fe in slot t: Similar to the inelastic system before the algorithm has the opportunity case, we also define xe[t]:=(xe[t])feЄFe to be the to converge. We also present extensive vector of elastic flow rates in slot t; and simulations to demonstrate the interaction between the two types of flows under our proposed algorithms. In particular, we show that due to the dynamic nature of the load balancing to denote the total number of elastic packets on mechanism implemented by inelastic sources, the link l. Associated with each elastic flow fe there elastic flows are able to push inelastic traffic onto exists a utility function Ue(.).that measures the less loaded routes and achieve higher rates. We “satisfaction” of that flow as a function of its show that this interaction maximizes the sum of mean injection the utilities of the elastic flows while satisfying rate the demands of the inelastic flows. We also In the text, we use x[t] := (xi[t],xe[t]) to denote compare the delay performance of our algorithm the vector of inelastic and elastic packets injected with and without the virtual queue into the network in slot t: Next, we provide a set implementation and illustrate that the virtual of assumptions to be used later in the analysis: queue scheme can reduce the end-to-end delays Assumption 1:The elastic routing matrix significantly. [Re]feЄFe has full row rank, which guarantees II OBJECTIVES: Inelastic Flow: We let fi denote that given q; there exists a unique p such that q =( an inelastic flow in the network with source si [Re]feЄFe )Tp. and destination di. Each inelastic flow fi is Assumption 2:The inelastic arrival process associated with a fixed set of routes Ri. The rth {Ai[t]}fi Є Fi is such that there exists a vector xi route of this set is described by a vector Ri(r) such that Ri(r) [l] = 1.if link lЄL is on that route, and zero otherwise. Let xi(r)[t] be the number of injected packets on the rth route of flow fi at time satisfying and slot t, and let xi[t] := (x i (r)[t]R(r) ЄRi fiЄFi be the vector of inelastic flow packets injected on each route in slot t: Note that we slightly abuse This condition implies that the inelastic flows are our notation by using xi to denote rate vector of supportable by the network, i.e., there exists a all inelastic flows, while i(r)stands for the rate of rate division of the inelastic flow rates over their flow fiЄFi over route rЄRi. We assume that the available routes which can support the arriving packet arrivals of the inelastic flow fi follow a traffic. stochastic process Ai[t] that is identically and Assumption 3: The utility functions {Ue(xe)}fe independently distributed (i.i.d.) over time with a are strictly concave, twice differentiable, and fixed mean rate, denoted by ai := E(Ai[t]), and a increasing functions. Such an assumption is finite second moment, i.e. E(A2i [t]) < ∞.To commonly used to capture the diminishing clarify the difference between Ai[t] and (x i (r) August Issue Page 49 of 68 ISSN 2229 5208 International Journal of Computer Information Systems, Vol. 5, No.2, 2012 returns to the elastic flows of an increase in the evolution in (1) which possess a more tractable service rate. and cleaner form. Assumption 4: For each elastic flow feЄFe, its Definition 1 (Stability): We say that a queue qR utility function Ue(x) satisfies: for each m>0 and is stable if MЄ[m, ∞],there exists and with (2) , ,satisfying where B is some finite positive value. We say that the network is stable if all aggregate queues for all xЄ[m,M].We note that Assumption 1 is {qR} for both inelastic and elastic flows are not critical in the proof of stability but will stable. simplify our proof. Also note that Assumptions 3 III. JOINT CONGESTION CONTROL AND and 4 on the utility functions are not restrictive LOAD BALANCING Stochastic Network and hold for the following class of utility Optimization (SNO) Problem: functions U(x)=wx(1-a) /(1-a),for a>0,which is known to characterize a large class of fairness concepts such as max-min fairness and weighted (3) proportional fairness ([25] and the references s.t. The Queue Evolution as in(1),Network therein).In subsequent discussions, when the Stability as in(2), distinction between real and non-real-time routes is unnecessary, we will simply refer to a route as R without any subscripts. Furthermore, for simplicity, we will use zl[t] for zl(xi[t]) and yl[t] A. Heuristic Fluid Model for yl(xe[t]).Queueing Architecture and In the fluid model scenario, all the dynamics and Evolution: In our system, for each link (I,j) Є L; a randomness are ignored, and the stochastic single priority queue is maintained at the constraints are replaced with static constraints. In transmitting node i, which holds all the packets particular, the inelastic flow fi is assumed to have whose routes traverse (I,j): Since the inelastic a fixed arrival rate ai, and the network stability flows are expected to have more stringent delay condition is replaced by a condition on total link constraints, their packets are always stored ahead rate being no more than capacity. Then, the SNO of those of the elastic flows. We let pl[t] denote problem reduces to the following problem in this the queue length of the buffer associated with link scenario. l at the beginning of slot t, and define Fluid Network Optimization (FNO) Problem: to be the total queue length on route R. Notice that pl[t] and qR[t] counts both the inelastic and elastic flows’ packets. During each time slot, the queue pl evolves as where x+ = max(0, x). This evolution is based on In our discussion, we will abbreviate the aggregate elastic and inelastic rates, yl(xe) and a link centric decomposition ([17]) and implicitly zl(xi); with yl and zl for brevity. Thus, the assumes that packets injected into the source optimization problem is to maximize the sum of nodes by the flows, denoted by x[t], arrive at the utilities of elastic flows when guaranteeing that downstream nodes instantaneously. In reality, inelastic flows are supported. It is not difficult to packets will reach downstream nodes only after a show that the optimum value of FNO is an upper queueing and propagation delay incurred in the bound for the optimum value of SNO. To see intermediate nodes. It is shown in prior works this, note that any solution {x(t)}t≥0 that solve ([27], [5], [28], [17]) that the inclusion of these dynamics do not affect the long-term stability and SNO must also satisfy where fairness characteristics of the system, and can be added to our queueing architecture by introducing a regulator queue before the queues associated and is defined similarly. Otherwise, the queue l with each link. Thus, in this work we use the cannot be stable. This is equivalent to condition (4) in FNO. Thus, FNO contains all the feasible August Issue Page 50 of 68 ISSN 2229 5208 International Journal of Computer Information Systems, Vol. 5, No.2, 2012 points of SNO. In Section III-B, we will design an algorithm under which SNO can get arbitrarily close to the FNO solution, and thus guarantees the optimality of SNO. We start by showing that This lemma implies that considering an inelastic there exists a unique Xe={xe}feЄFe that solves flow fi,all routes in the optimal solution with a the FNO problem under Assumption 2 and 3 1. positive flow have the same value of ß. We note Proposition 1: If Assumption 2 and 3 hold, then that αl of FNO is closely associated with the the X*e={xe}feЄFe that solves the network queue length pl of SNO, and correspondingly βR optimization problem is unique. of FNO is closely associated with the aggregate Proof: The optimization problem has a unique queue length on a route qR of SNO. Such solution because the utility functions are strictly connections are revealed and exploited in several concave, and constraints (4) and (5) are linear. To earlier works for designing different network solve the FNO problem, we construct a partial algorithms (e.g. [15], [16], [6], [7], [26]). Joint Lagrangian. Define αl to be the Lagrange Congestion Control and Load Balancing multipliers associated with constraint (4). Then, Algorithm for the FNO problem: the partial Lagrangian can be written as 1.Queue evolution for link l: , Where (v(t))+p(t) is zero if v(T)<0and P(t)=0;and v(t) otherwise. Congestion Controller for elastic flow fe: and Since FNO problem satisfies Slater’s condition Load Balancing ([4]) due to Assumption 2, the strong duality implemented for inelastic flow fi: holds. We can then conclude that there exists x* :=(x*e,x*i),and α*:=(αl)l 1.x* solves FNO problem; and 2.x*Є arg (8) maxx≥0 L(xi,xe,α*).Note that and Where satisfies - (9) where βR := ΣlЄLR[l] αl. This decomposition and suggests that (i) The elastic flow fe should allocate its rates such that when the system reaches the equilibrium, we have xi(r)(t) = 0 for all r. This implies that qRt(r)(t)= (t) fpr xi(r)(t)>0 and qRt(r)(t)>= (6) for xi(r)(t)=0 . Thus, at the equilibrium point, (ii) The inelastic flow fi, should distribute its qRt(r)(t) satisfies Lemma 1. Furthermore, from (9), it is easy to see that packets over its available routes such that for all t (10) (7) Next, we will show the stability and optimality of our joint congestion control and load balancing Since the optimization problem (7) has a linear algorithm. objective,the following lemma holds ([4]): Proposition 2: Under Assumption 1, 2 and 3, the Lemma 1: For any R(r)iЄRi, we have: joint congestion control and load balancing algorithm is globally asymptotically stable, i.e. and limt→∞xe(t) = x*e starting from any x(0); where August Issue Page 51 of 68 ISSN 2229 5208 International Journal of Computer Information Systems, Vol. 5, No.2, 2012 x*e is the optimal solution to the FNO problem. congestion controller is inclined to inject more Furthermore, (10) holds. packets into the network with larger K. Also note B. Stochastic Model that the load balancing implementation is slightly We now return to the original SNO problem with different from the fluid model version to a minor variation: SNO Problem with Parameter accommodate the randomness in the arrival processes for inelastic flows. The update is modified to ensure that K (SNO-K): The Queue Evolution as in(1); Network Stability holds for all t: The next as in(2); proposition establishes the stability and optimality of the joint algorithm for the stochastic system. Proposition 3: Under Assumptions 1, 2, 3 and 4, the joint congestion control and load balancing algorithm stabilizes the system in the where K is a positive design parameter. We will sense that the Markov chain (p[t],xi[t]) is positive see that K parameter is critical in eliminating the recurrent with effect of randomness in the stochastic system on the long-term performance. Note that the solution to the SNO-K problem is independent of the value of K, and its optimum solution is identical to the solution of the SNO problem. Motivated by , the analysis in the fluid model, we propose the and guarantees that the rate allocation satisfies, following joint congestion control and load balancing algorithm: Joint Congestion Control and Load Balancing Algorithm for the SNO-K problem: 1. Queue evolution for a link l: Here x*e is the optimal solution to the SNO-K problem, is an arbitrarily chosen positive Congestion Controller for elastic flow fe: constant, and Є and ~c3 are positive values. IV. EXTENSION OF THE ALGORITHM A. Virtual Queue Algorithm where M is a positive constant satisfies Inelastic applications are delay sensitive, hence we assume that packets from inelastic flows have strict priority over their elastic counterparts. 2.Load Balancing implemented for inelastic flow Thus, the inelastic flows do not see the elastic fi: flows in the queues they traverse. But in some cases a link might be critically loaded by the inelastic traffic itself, thus resulting in large delays. Also, elastic traffic may have some delay or constraints that are non-negligible. An effective equivalently, way of reducing the experienced delay is by including virtual queues that are served at a fraction of the actual service rate, and by using the virtual queue-length values as prices ([12]). Where satisfies To that end, we introduce two types of virtual queues with parameters, ρ1 and ρ2, which control the total load and the inelastic flow load, and respectively. Here for simplicity, we go back to the fluid model to design and analyze the joint congestion control and load balancing algorithm using virtual queues. We would like to have Remark: The factor 1=K in the congestion control ~x*= solve the following optimization equation comes from the factor K in the problem. optimization problem. It can be interpreted as the FNO Problem with Virtual Queues (FNO-VQ): aggressiveness factor of the elastic flow, as the August Issue Page 52 of 68 ISSN 2229 5208 International Journal of Computer Information Systems, Vol. 5, No.2, 2012 (11) and and Remark: In the above algorithm, note that the congestion control algorithm only responds to the Where 0< ρ2<= ρ1<1 To guarantee the feasibility virtual queues for elastic flows, but the load of the optimization problem above, we replace balancing algorithm responds to both the virtual our earlier Assumption 2 with: Assumption 5: queues for elastic flow and inelastic flows. There exists a xi such that Further, the actual queue length is not used in the algorithm. In the following proposition, we show the equilibrium point of the algorithm with virtual queue is the optimal solution of (11). and Proposition 4: Under Assumptions 1, 3 and 5, the joint congestion control and load balancing algorithm for the FNO-VQ problem is globally To solve the FNO-VQ problem, we first asymptotically stable, i.e., = introduce virtual queues for elastic and inelastic where is the solution of the network flows on each link respectively. The virtual queue optimization length µl(t) for elastic flows evolutes as follows: Thus, we focus on the dynamics of file arrivals and departures, where we assume that (i)The number of inelastic flows are fixed. The virtual queue length for inelastic flows (ii) Files belonging to elastic flows dynamically evolves as follows: arrive and depart. We assume that files arrive according to independent Poisson processes with rate files/sec, the file sizes are independently Note that when the total instantaneous traffic load and exponentially distributed, with mean file size is larger than ρ1cl or the inelastic traffic load is bits. Further, let ne(t) denote the number of larger than ρ2cl, the virtual queues will build up, files belonging to flow fe in the network at time t. and the network controller will reduce the traffic The following result states that our joint load. Based on this virtual-queue scheme, we congestion control and load-balancing algorithm have the following joint congestion control and maximizes the network throughput region. load balancing algorithm:Joint Congestion Proposition 5: Under Assumptions 1-3, the Control and Load Balancing Algorithm for FNO- network is stable under the joint congestion VQ problem:Virtual queue evolution for a link l: control and load balancing algorithm if there Elastic exists such that flows: ; Inelasticflows: and for any . Congestion Controller for elastic flow . (12) whereis the aggregated virtual queue length of the elastic Under Assumptions 1, 4, and 5, the network is flow. 1. Load Balancing implemented for stable under the joint congestion control and load inelastic flow fi: balancing algorithm with virtual queues if there exists such that , Where satisfies August Issue Page 53 of 68 ISSN 2229 5208 International Journal of Computer Information Systems, Vol. 5, No.2, 2012 on the bottleneck link (2; 5). As a reaction to the increased contention from the elastic flow, the load balancing mechanism of the inelastic flows And for any we automatically pushes more and more traffic of the inelastic flow onto route 1.Note that although increasing the aggressiveness of the elastic flow will increase the utilization of the network, it will have result in more delays on the network flows as the queue length over the whole network grows, as shown by Proposition 3. Proposition 3 also suggests that larger K resulting better (13) convergence to the optimal operating point, which is confirmed in the above simulation. V. SIMULATION RESULTS In this section, we provide the simulation results B. More Complex Topology To illustrate other for our algorithms under the stochastic model facets of our algorithm, we conducted our where the arrival process of the inelastic flow fi is simulation in a more complicated network with such that Ai[t] is Poisson distributed with mean ai different flow assignments. The topology of the for each t: A.The Effect of the Aggressiveness of network is shown in Figure 5. The capacity of all the Inelastic Flow: We noted in Section III-B that the links in the network is 20, and we used elastic the factor K represents the ‘aggressiveness’ of the flows with identical utility functions and K = 200 elastic flows. Also, it is revealed in Proposition 3 in our simulation, expecting close to optimal that K can be used to control the proximity to the utilization (as shown in Figure 4). We simulate a optimal allocation. Here, we test these results for sequence of scenarios discussed in five phases. In the case of proportionally fair allocation, which Phase 1, two disjoint inelastic flows with the corresponds to having the utility function is routes as shown in Figure 5 share the network, chosen as ([25]): Ue(x) = αln x, thus having rates ai1 = 20 and ai2 = 10. The average rates provided on each route by our joint algorithm are given in Figure 5. When the two inelastic flows share a common bottleneck link in and In this first set of Phase 2, the load balancing algorithm will shift simulations, we considered the network shown in part of the traffic from the bottleneck link to yield Figure 3 with the indicated link capacities and the average rates given in Figure 6. inelastic and elastic flows. Note that the arrival rate of the inelastic flow is ai = 15 to be distributed over Fig. 5. Phase 1: Two inelastic flows Fig.6. Phase 2: Two inelastic flows with disjoint routes. with intersecting routes. In Phase 3, an elastic flow enters the system and the two dashed routes.The joint shares a link with fi2 as in Figure 7. We can see fig 3:topology of network from the average rates given in the figure that this fig 4:effect of k on each route elastic flow not only has an effect on fi2 but also algorithm for the SNO-K problem is shifts the rate of fi1. Here, it can be seen that the implemented for this network and the mean interaction between the flows becomes complex elastic rate allocation is computed for different even for small networks, and it is not clear what values of K: Figure 4 illustrates the effect` . of K the best allocation is. Yet, through our joint on the rates of the elastic flow and the algorithm, fe1 is able to operate dynamically distribution of the inelastic flow’s rate over its close to the full capacity of all the resources available routes. We see that as the elastic flow available to it. After adding another elastic flow becomes more aggressive, it achieves a higher fe2 into the network which is disjoint with all throughput and thus consumes greater resource other flows in Phase 4 shown in Figure 8, we can August Issue Page 54 of 68 ISSN 2229 5208 International Journal of Computer Information Systems, Vol. 5, No.2, 2012 see that it has no effect on the rates of all other implementation on delay. The simulation is routes, and it fully utilizes that route. conducted in the network showing in Figure 11. The parameter ρ1was set to 0.95 and ρ2 was set to 0.9 over all links. Table 1 Fig. 7. Phase 3: Two intersecting inelastic flows, fig8:phase 4:2 intersecting inelastic and one elastic flow that interacts with them.Flows and 2 with disjoint routes. In Phase 5, a third elastic flow fe3 enters and shares common links with both fi1 and fi2, as shown in Figure 9. We can see that since fe1 also shares links with fi2, fe3 also has effect on it. It can be easily verified that xe1 = xe3 = 15 is the optimal operating point, and the average rates achieved by our algorithm is very close to optimal as predicted by Proposition 3. To study the importance of dynamic load balancing, we also (11) simulated a static rate distribution algorithm as a basis for comparison. This algorithm equally splits the inelastic traffic onto each of its routes Fig. 11. Network topology (assume it is feasible in the network), and does TABLE I:rate and delay on each routes the congestion control of the elastic flows in the same manner as in our algorithm. This algorithm RESULT & ANALYSIS: is implemented for the scenario in Phase 5 with the average rates indicated in Figure 10. We see that due to the absence of dynamic load balancing, the elastic flows cannot utilize the network fully since the rates assigned to the inelastic flows are fixed.Under the logarithm utility function, this approach achieves a utility of 1:61αK while our algorithm achieves 2:71αK on the elastic flow fe3. Fig. 9. Phase 5: A third elastic flow enters Fig. 10. Performance under static that intersects with two inelastic flows. rate distribution for inelastic traffic. C. Simulation using the Virtual Queue Algorithm In this simulation, we use the joint congestion Fig:12 throughput of network with different control and load balancing algorithm with virtual protocols queue to show the impact of the virtual queue August Issue Page 55 of 68 ISSN 2229 5208 International Journal of Computer Information Systems, Vol. 5, No.2, 2012 Table I and fig12 compares the performance with and without virtual queues. As we can REFERENCES observe, under the original algorithm, Route 1 is [1] E. Altman, T. 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In Proceedings of International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOPT), 2005. AUTHORS PROFILE 1].A.RARAJU,Received B.Tech degree from Sri Venkateshwara University,Trupathi.and M.Tech in Computer Science & Engineering from Kottam college of engineering, Kurnool,JNTU Ananthapur, Andhra Pradesh, India. raraju22@gmail.com 2]. K.SRINIVAS, received MCA degree from Osmania University and M.Tech in Computer Science & Engineering from JNTU Hyderabad, and Ph.D from Rayalaseema University(submitted), Andhra Pradesh, India. And also CISCO CCNA Certified. Currently he is Professor and HOD Of Computer Science Department in Kottam College of Engineering, Kurnool, AP, India. kipgs2008@gmail.com August Issue Page 57 of 68 ISSN 2229 5208