Near-optimal Delay Constrained Routing in Virtual Circuit Networks

Transcription

1 Near-otima Deay Constrained Routing in Virtua Circuit Netorks Hong-Hsu Yen and Frank Yeong-Sung Lin Deartment of Information Management Nationa Taian University Taiei, Taian, R.O.C. Te: Fa: Emai: Abstract -- An essentia issue in designing, oerating and managing a modern netork is to assure end-to-end Quaity-of- Service (QoS) from users ersective, and in the meantime to otimize a certain average erformance objective from the system s ersective. In this aer, e consider the robem of minimizing the average cross-netork acket deay in virtua circuit netorks subject to an end-to-end deay constraint for each origin-destination user air. The robem is formuated as a muticommodity netork fo robem ith integer routing decision variabes, here additiona end-to-end deay constraints are considered. The difficuties of this robem resut from the integraity nature and articuary the nonconveity associated ith the end-to-end deay constraints. The basic aroach to the agorithm deveoment is Lagrangean reaation in conjunction ith a number of otimization-based heuristics. In the comutationa eeriments, it is shon that the roosed agorithm cacuates soutions hich are ithin % and % of otima soutions under ighty and heaviy oaded conditions, resectivey, in minutes of CPU time for netorks ith u to 6 nodes. Inde Terms -- Otimization, Lagrangean Reaation, End-toend QoS, Routing Assignment. INTRODUCTION To ensure user-erceived end-to-end QoS requirement is one of the most imortant issues in roviding modern netork services, hich tyicay requires sohisticated design of routing and caacity management oicies. User-erceived end-to-end QoS measures incude, for eame, mean acket deay, acket deay jitter and acket ost robabiity. Besides users ersective of QoS, from the service roviders ersective (hich is a traditiona vie of netork erformance management), otimizing a certain system-eve erformance measure, e.g. overa netork utiization or average cross-netork deay among a users, is another major concern. Unfortunatey, these to ersectives/objectives may not be entirey agreeabe ith each other. This then aces a major chaenge to netork managers and therefore cas for an integrated methodoogy to consider these to ersectives in a joint fashion. The routing robem in virtua circuit netorks has been a traditiona research toic in comuter netorks and has attracted even more attention since the emergence of the Asynchronous Transfer Mode (ATM) technoogy. Hoever, most revious research on virtua circuit routing considers the objective function of minimizing the average end-to-end acket deay [,,5], hich addresses a system-otimization ersective ithout taking individua users into account. In [], Cheng and Lin took a user-otimization aroach and considered a fairness issue by minimizing the maimum individua end-to-end acket deay in virtua circuit netorks. In this aer, e attemt to jointy consider both ersectives. More recisey, e consider the virtua circuit routing robem of minimizing the average acket deay subject to end-to-end acket deay constraints for users. This robem is a difficut NP-comete robem as indicated in [6]. An otimizationbased aroach is then devised to attack the robem, here the robem is formuated as a mathematica rogramming robem, fooed by roosing an agorithm based on Lagrangean reaation. It is shon in the comutationa eeriments that the roosed agorithm is both efficient and effective. The remainder of this aer is organized as foos. In Section, a mathematica formuation of the routing robem is roosed. In Section, a soution aroach to the routing robem based on Lagrangean reaation is resented. In Section, heuristics are deveoed to cacuate good rima feasibe soutions. In Section 5, comutationa resuts are reorted. Finay, Section 6 concudes this aer.. PROBLEM FORMULATION The virtua circuit netork is modeed as grah here the rocessors are deicted as nodes and the communication channes are deicted as arcs. We sho the definition of the fooing notation. V ={,,,N}, the set of nodes in the grah L the set of communication inks in the communication netork W the set of Origin-Destination (O-D) airs in the netork Corresondence Author

2 γ C P δ g D (ackets/sec): the arriva rate of ne traffic for each O-D air, hich is modeed as a Poisson rocess for iustration urose (ackets/sec), the caacity of each ink L a given set of sime directed aths from the origin to the destination of O-D air a routing decision variabe hich is hen ath P is used to transmit the ackets for O-D air and otherise the indicator function hich is if ink is on ath and otherise the aggregate fo over ink, hich is equa to γ δ P the maimum aoabe end-to-end deay for O-D air Under the assumtion of Keinrock s indeendent assumtion [], the arriva of ackets to each buffer is a Poisson rocess here the rate is the aggregate fo over the outbound ink. Assume that the transmission time for each acket is eonentiay distributed ith mean C -. Hence, refer to revious research [,, 5], each buffer is modeed as an M/M/ queue. It is remarkabe to address that the formuation can be etended to any non M/M/ mode ith monotonicay increasing and conveity erformance metrics. For the iustration urose, the formuation i be based on the M/M/ mode. To determine a ath for each O-D air to minimize the average acket deay ith maimum aoabe end-to-end transmission deay is formuated as a noninear combinatoria otimization robem, as shon beo. Z IP g " = min (IP ) γ C g L P g = P P δ C g L D W (.) γ δ C L (.) = W (.) = or P, W. (.) Constraint (.) requires that the end-to-end acket deay shoud be no arger than D for each O-D air. Constraint (.) requires that the aggregate fo on each ink shoud not eceed the ink caacity. Constraints (.) and (.) require that the a the traffic for each O-D air shoud be transmitted over eacty one ath. The above formuation is a noninear muticommodity fo robem, since each O-D air transmits different tye of traffic over the netork. And it is easy to sho that (IP ) is a nonconve rogramming robem by δ verifying the Hessian of ith resect to. C g L P For the urose of aying Lagrangean reaation method, e transform the above robem formuation (IP ) into an equivaent formuation (IP). In (IP), to auiiary variabes are introduced: y is defined as δ and f denotes the P estimate of the aggregate fo. Z IP = min y L P γ D L f (IP) W (.) = W (.) = or P δ y P, W (.) W, L (.) y = or W, L (.5) g f C L (.6) f L. (.) Redundant constraints associated ith these auiiary variabes from (.) to (.) are added. Note that Constraints (.) and (.6) shoud be equaities, and it is cear that the equaity shoud hod at the otima oint. By introducing these auiiary variabes, the Lagrangean reaation robem can be decomosed into indeendent and easiy sovabe subrobems.. LAGRANGEAN RELAXATION The agorithm deveoment is based uon Lagrangean reaation. We duaize Constraints (.), (.) and (.6) to obtain the fooing Lagrangean reaation robem (LR). = f + y Z D ( t, min[ t ( γ C f ) L D + v δ y ) + P L ( P L L u ( g f )] (LR) = W (.) = or P, W (.) y = or W, L (.) f L. (.) C We can decomose (LR) into to indeendent subrobems.

3 Subrobem : for min ( v + uγ ) δ (SUB) P L subject to (.) and (.). Subrobem : for y and f t y f min[ ( + γ v y u f ) L D t ] (SUB) subject to (.) and (.). (SUB) can be further decomosed into W indeendent shortest ath robem ith nonnegative arc eights. It can be easiy soved by the Dijkstra s agorithm. The D t tem in the objective function of (SUB) can be droed first and added back to the objective vaue since it i not affect the otima soution of (SUB). Then (SUB) can be decomosed into L indeendent subrobems. For each ink L min[ γ f * + t y u y = or W and f C. f v ] y (SUB.) (SUB.) is a comicated robem due to the couing of f y and f. On the other hand, the * term in the γ objective function of (SUB.) is a nonnegative and monotonicay increasing function ith resect to f, and it i not affect the otima vaue of the fooing terms in the (SUB.). Therefore, the agorithm deveoed in [] can be used to sove (SUB.). Hence, the agorithm to sove (SUB.) is as foos: t Ste. Sove ( v = ) for each O-D air, ca C f them the break oints of f. Ste. Sorting these break oints and denoted as f, f,.., n f Ste. At each interva, t v i f f i+ f and is otherise., y (f ) is if Ste. Within the interva, f f, et a be t y ( f ) and b be v y ( f ), then the i f i+ oca minima is either at the boundary oint, i+ f, or at oint * f = C a. u f i or Ste 5. The goba minimum oint can be found by comaring these oca minimum oints. According to the agorithms roosed above, e coud successfuy sove the Lagrangean reaation robem otimay. By using the eak Lagrangean duaity theorem (for any given set of nonnegative mutiiers, the otima objective function vaue of the corresonding Lagrangean reaation robem is a oer bound on the otima objective function vaue of the rima robem), Z D (t, is a oer bound on Z IP. We construct the fooing dua robem to cacuate the tightest oer bound and sove the dua robem by using the subgradient method. Z D = ma Z D ( t, t, v. (D) Let the vector S be a subgradient of Z D ( t, at (t,. In iteration of the subgradient otimization rocedure, the S mutiier vector m =(t,u,v ) is udated by m + = m + α, here S y ( t, = ( D, g f, C L f δ P y ). The ste size α is determined by h Z IP Z D ( m ) δ, S h here Z IP is an rima objective function vaue (an uer bound on otima rima objective function vaue), and δ is a constant ( δ ).. GETTING PRIMAL FEASIBLE SOLUTIONS To obtain the rima soutions to the minimized average acket deay ith maimum aoabe end-to-end deay constraints robems, soutions to the Lagrangean reaation robems (LR) is considered. For eame, if a soution to (LR) is aso feasibe to (IP), i.e., satisfy the caacity constraints and end-to-end deay constraints, then it is considered as a rima feasibe soution to (IP); otherise, it i be modified so that it may be feasibe to (IP). Three getting rima heuristics are deveoed to imrove the effectiveness of the agorithms. For eame, hen a soution to (LR) is found, the routing assignments for the maimum end-to-end deay ath is reassigned to another ath to reduce the vaue of maimum end-to-end deay. For another case, consider that the end-to-end deay constraints, hen the endto-end deay constraints is vioated, identify the aths that vioate end-to-end deay constraints, the arc eights aong these aths are increased, then the routing assignments are recacuated. On the other hand, consider the caacity constraints, hen a soution to (LR) is infeasibe for caacity

4 constraints, the arc eight for the overfo ink is increased, and then the routing assignments are recacuated. According to the comutationa eeriments, the second heuristic can get a better soution in many cases. 5. COMPUTATIONAL EXPERIMENTS The comutationa eeriments for the agorithms deveoed in Sections and are coded in C and erformed on a PC ith INTEL TM PII- CPU. We tested the agorithm for netorks --,, ith, and 6 nodes. The netork tooogies are shon in Fig., and. 5 Figure. -node 5-ink Netork Figure. -node 5-ink Netork Figure. 6-node 6-ink Netork The maimum number of iterations for the roosed dua Lagrangean agorithm is, and the imrovement counter is. The ste size for the dua Lagrangean agorithm is initiaized to be and be haved of its vaue hen the objective vaue of the dua agorithm, does not imrove for iterations. It is assumed that the traffic demand of each O-D air is one acket er second. Unike others ork in [], the candidate ath set does not need to be reared in advance and a ossibe candidate aths are considered for each O-D air. We erform to sets of comutationa eeriments. In the first set of comutationa eeriments, the choice of the D vaue is fied as to eamine the soution quaity of the minimum average acket deay robem. Tabe summarizes the resuts. The first coumn is the tye of the netork tooogy. The second coumn is the ink caacity. The third coumn is the maimum aoabe end-toend deay (D ). The forth coumn reorts the oer bound of the roosed dua Lagrangean robem. The fifth coumn reorts the uer bound of the roosed dua agorithm. The sith coumn reorts the error ga beteen the oer bound and the uer bound. The seventh coumn reorts the maimum end-to-end deay among a O-D airs. As can be seen in the sith coumn, the ga beteen the oer bound and the uer bound are very tight for a different netork tooogies and ink caacities hen the vaue of D is oose as comared to the maimum end-to-end deay among a O-D airs. Since the vaue for the maimum aoabe end-to-end deay (D ) have a significant imact on the soution of the minimum average acket deay robem. In the second set of comutationa eeriments, e try to eamine the imact of the D vaue on the soution quaity of minimum average deay. Fig., 5 and 6 shos the resuts for the,, netork. It is cear to see that the uer bound remains amost the same ith different D vaue. When the D vaue beo a certain threshod (as indicated in third coumn of Tabe ), the rima soution coud not be found. Tabe summarizes this resut. The first coumn is the tye of the netork tooogy. The second coumn is the ink caacity. The third coumn is the threshod of the maimum aoabe end-to-end deay (D ). The forth coumn reorts the oer bound of the roosed dua Lagrangean robem. The fifth coumn reorts the uer bound of the roosed dua agorithm. The sith coumn reorts the error ga beteen the oer bound and the uer bound. The seventh coumn reorts the maimum end-to-end deay among a O-D airs. The eighth coumn reorts the resuts from []. The maimum end-to-end deay vaue in the seventh coumn imies the otima vaue for the minima End-to-end Deay Routing robem deveoed in []. There is one thing that needs to be addressed: a ossibe candidate aths are considered for each O-D air in this aer but ony three candidate aths are re-chosen for each O-D air in []. Athough e obtain a tighter uer bound than the minima end-to-end deay routing robem deveoed in [], this comarison is not on the same basis. On the other hand, the ga beteen the oer bound and the uer bound of the minimum average deay robem is sti very tight, hich indicate that the agorithms that e deveoed can achieve good system objective (average acket deay) even in stringent end-to-end deay requirements. 6. CONCLUSIONS As comared to the ork in [], hich tried to achieve better fairness among users by minimizing the maimum endto-end deay for virtua circuit netorks ithout considering the system ersective (minimize the average acket deay). In this aer, for the first time, e considered the robem of minimizing the average acket deay ith maimum aoabe end-to-end deay requirements, hich indicate that e try to obtain good system erformance under user s end-to-end deay requirements. We formuate this robem as a nonconve and noninear

5 muticommodity integra fo robem. The nonconve and discrete (integer constraints) roerties make the robem very difficut. We take an otimization-based aroach by aying the Lagrangean reaation technique in the agorithm deveoment. According to the comutationa eeriments, the error ga beteen the uer bound and the oer bound is so tight that e can caim that a near otima soution is found. When the maimum end-to-end deay requirements are more and more cose to the threshod, the uer bound (average acket deay) remains amost the same, this indicate that this soution aroach can obtain good average acket deay soution under stringent end-to-end QoS requirements. REFERENCES [] B. Gavish and S. L. Hanter, An Agorithm for Otima Route Seection in SNA Netorks, IEEE Trans. on Comm., COM-,. 5-6, Oct. 98. [] K. T. Cheng and F. Y. S. Lin, Minima End-to-End Deay Routing and Caacity Assignment for Virtua Circuit Netorks, Proc. IEEE Gobecom,. -8, 995. [] F. Y. S. Lin and J. R. Yee, A Ne Mutiier Adjustment Procedure for the Distributed Comutation of Routing Assignments in Virtua Circuit Data Netorks, ORSA Journa on Comuting, Vo., No., Summer 99. [] L. Keinrock, Queueing Systems, Wiey-Interscience, Ne York, Vo. and, [5] P. J. Courtois and P. Sema, An Agorithm for the Otimization of Nonbifurcated Fos in Comuter Communication Netorks, Performance Evauation,. 9-5, 98. [6] S. Chen and K. Nahrstedt, An Overvie of Quaity of Service Routing for Net-Generation High-Seed Netorks: Probems and Soutions, IEEE Netork,. 6-9, Nov./Dec

6 Netork tooogy Link caacity TABLE COMPARISON OF SOLUTION QUALITY OBTAINED BY VARIOUS NETWORKS D Loer bound Uer bound Error ga (%) Maimum end-to-end deay TABLE COMPARISON OF SOLUTION QUALITY OBTAINED BY VARIOUS NETWORKS AT THE THRESHOLD OF MAXIMUM ALLOWABLE END-TO-END DELAY Netork tooogy Link caacity Threshod of D Loer bound REQUIREMENTS Uer bound Error ga (%) Maimum end-to-end deay Resuts from [] : The ork in [] did not erform the comutationa eeriments in these netork settings zh(msecs) netork(uer bound vs. maimum aoabe end-to-end deay vaue) D (msecs) Figure. Uer bound for different D vaue in netork 6

7 zh(msecs) netork(uer bound vs. maimum aoabe end-to-end deay vaue) 5 6 D (msecs) Figure 5. Uer bound for different D vaue in netork netork(uer bound vs. maimum aoabe end-to-end deay vaue) zh(msecs) D (msecs) Figure 6. Uer bound for different D vaue in netork