Aswakul, C

Barria, J

IEEE Communications Letters

C. Aswakul

J. Barria

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Performance analysis of single-link system with nonlinear equivalent capacity

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IEEE COMMUNICATION LETTERS, VOL. 4, NO. 2, FEBRUARY 2000 71Performance Analysis of Single-Link System withNonlinear Equivalent CapacityChaodit Aswakul and Javier BarriaAbstract—This letter presents a procedure to estimate theperformance of a single-link system with nonlinear equivalentcapacity. The idea is to convert the problems from thenonlineardomain into the linear domain so that the efficient numericaltechniques in the linear domain can be employed. Based on theKaufman–Roberts’s recursion [4], [6], the proposed procedure hasthe time complexity = ( ) and the space complexity= ( ),where is the link capacity and is the number of call types.The absolute accuracy [5, Sec. 4.3] of the proposed procedure inestimating the blocking probabilities is in the order of10 3–10 2for the reported results.Index Terms—Asynchronous transfer mode, equivalent ca-pacity, single-link system.I. INTRODUCTIONI N CONVENTIONAL traffic engineering, various mathe-matical models and approximation techniques have beenproposed (e.g., [8]) and successfully employed for performanceanalysis of conventional systems such as circuit switching. Allthese mathematical tools rely on an implicit assumption aboutthe linear relationship between the number of connectionsand the amount of link capacity required for theconnections. In this letter, function will be referred to astheequivalent capacity.1A linearequivalent capacity (i.e., ) can be a reason-able assumption when analyzing conventional systems. How-ever, this linear relationship may not hold in cases such as asyn-chronous transfer mode (ATM) networks. For ATM networks,cell streams from different connections arestatistically multi-plexedin order to improve the utilization of ATM multiplexers.It has been found that the effect of this statistical multiplexingcan be well captured by an equivalent capacity that in-creases monotonically with decreasing slope asincreases, i.e.,it is of nonlinearform [2], [3].This letter is thus concerned with the performance analysis inthe nonlineardomain of equivalent capacity. In particular, weconsider a single finite-capacity ATM link accessed by manycall streams. This system is assumed to have aproduct-form so-Manuscript received March 22, 1999. The associate editor coordinating thereview of this letter and approving it for publication was .Dr. K. Sriram. Thework of C. Aswakul was supported by Anandamahidol Foundation, Thailand.The authors are with the Department of Electrical and Electronic Engineering,Imperial College of Science, Technology and Medicine, University of London,London, SW7 2BT, U.K. (e-mail: c.aswakul@ic.ac.uk; j.barria@ic.ac.uk).Publisher Item Identifier S 1089-7798(00)01249-7.1The terminology “equivalent capacity” was first introduced in [3] underthe situation of a first-in-first-out ATM multiplexer accessed by 2-state fluidsources. However, we prefer using this terminology in a wider sense to generallydefinethe amount of link capacity required for a given number of connections.lution and thecomplete-sharingcall admission control (CAC)scheme. The probability of call blocking is considered as themain performance measure of interest [4]–[8]. To date, althoughthe blocking probability expression is simple and well-known,there is no efficient recursive algorithm available that can cal-culate the blocking probability, except for the ordinary cases inwhich is a linear function of . Thus, to solve problems in thenonlineardomain, we propose here to first convert the problemsinto thelinear domain so that the efficient numerical techniquesin the linear domain (e.g., the Kaufman–Roberts’s recursion [4],[6]) can be employed.II. M ATHEMATICAL MODELConsider a link with capacity and call types. Let indexthe call type ( ). Type- calls arrive as aPoissonstream of rate and as varies it, indexesindependentPoissonstreams.Call holding timesare independent of each other andthey are independent of call arrivals. The holding times of type-calls are identically distributed with mean . Denote theprobability of blocking type-calls by .Let be the equivalent capacity of type-calls. That is,type- connections require link capacity of magnitude ).Define a link-state vector ), where isthe number of type-connections. Thecomplete-sharingCACscheme admits a call as long as, after taking into account thenew connection, is still contained in the state space:(1)where is the set of nonnegative integers.Let be the number of type-connections on the linkat time . The stochastic processthen has steady state distribution, , givenby aproduct-form solution[6](2)(3)where ) is theoffered loadof type- call stream.And, since the call arrival streams are Poisson,(4)1089–7798/00$10.00 © 2000 IEEE72 IEEE COMMUNICATION LETTERS, VOL. 4, NO. 2, FEBRUARY 2000where is the -dimensional vector consisting of only zerosexcept for a one in theth component. Note that the form ofequivalent capacity influences the blocking probabilities via theexpression of the state space.III. PROPOSEDAPPROXIMATION PROCEDUREConsider the stochastic process representing thelink capacity utilized by type-calls at time . Defineas the steady-state marginal distribution of :(5)Since typically takes its peak around ,we propose here to approximate the nonlinear function bya linear function such that(6)where denotes and denotes . Thatis, the proposed approximation fits to a linear functionat the two nearest integer points about the mean. This linearfunction can be written straightforwardly as(7)where(8)(9)Now, replacing in (1) by from (7), wethen have the approximated state space(10)where(11)From (10) and (11), we may interpret as the slope of alinear equivalent capacity and as amodifiedlink capacity.Thus, (7)–(11) convert our problem from thenonlineardomaininto thelinear domain and we shall refer to this linearly approx-imated problem aslinear approximation model.To solve forthe blocking probability in thelinear approximation model,wecan apply any numerical techniques in thelinear domain (e.g.,[4], [6]). However, to formulate thelinear approximation modelabove, we assume the knowledge of . Thisis not trivial because is the carried load of type-callstream, i.e.,(12)and is actually the output parameter to be computed. Hence,the linear approximation model cannot be formulated directly.To resolve this problem, afixed-point iteration(e.g., [1]) can beapplied as follows.Fig. 1. Equivalent capacity oftraffic 1 andtraffic 2.Fig. 2. Blocking probability oftraffic 1.Consider the blocking probability as the recursive parameter.Let represent the value of at the th iteration forand let be the initial value for the re-cursion. Then, the fixed-point iteration procedure takes the fol-lowing steps.a) Define .b) Calculate for all by (12) with replaced by.c) Calculate and for alld) Calculate the parameters in the linear approximationmodel (i.e., and ) by (8), (9), and (11).e) Calculate for all in the linear approximation modelby the Kaufman–Roberts’s recursion [4], [6].f) Update for all .g) Iftolerance parameterfor all , then stop the recursion.Otherwise, set and go back to step b).From our numerical experience, this recursion always con-verges within a few iterations (generally less than 5). Thus, itscomputational complexity is in the same order as that of theASWAKUL AND BARRIA: PERFORMANCE ANALYSIS OF SINGLE-LINK SYSTEM 73Fig. 3. Blocking probability oftraffic 2.Kaufman–Roberts’s recursion. That is, the time complexity isand the space complexity is .IV. NUMERICAL RESULTSAs an example, consider two call types. Suppose that type-1call stream (traffic 1) and type-2 call stream (traffic 2) have theequivalent capacity as depicted in Fig. 1. The form ofis adopted from [3] where in themost (least) nonlinear case. The values of in megabitsper second (Mb/s) fort affic 1 are (3,0), (2,7.0711), (1,14.1421)and (0,21.2132) in Cases 1–4, respectively. The values ofin Mb/s for traffic 2 are (0,21.2132) in all the cases.Figs. 2 and 3 depict the blocking probabilities computed fromthe linear approximation model, in comparison with their exactvalues. The exact values are obtained from directly computingthe product-form solution and summing over the blocking-stateoccupancy probabilities. The offered loads oftraffic 1 andtraffic 2 are equally increased and the link capacity is fixedat 150 Mb/s. The figures indicate a good matching betweenthe approximate values and the exact values over a practicalrange of offered loads. The absolute accuracy [5, Sec. 4.3] iscalculated, , where denotes theapproximation for and is the number of experiments. ForCases 1–4, , respec-tively. That is, increases as the equivalent capacity is morenonlinear but it is still in an acceptable order ( – .V. CONCLUSIONSAn approximation procedure, termedlinear approximationmodel, is proposed in this letter to analyze a complete-sharingsingle-link system withnonlinear equivalent capacity. Whenemployed with the Kaufman–Roberts’s recursion [4], [6], thelinear approximation modelrequires the time complexity =and the space complexity = . Based on the reportedresults, the absolute accuracy of the proposed procedure is inthe acceptable order of – . Due to simplicity of theproposed approximation, one might expect that the degree ofits accuracy could be vulnerable to extreme cases where theequivalent capacity is highly nonlinear. Although this highnonlinearity may not happen in most of practical cases, weare at present working on obtaining theoretical bounds for theaccuracy of the procedure proposed here and the results will bereported in a forthcoming paper.REFERENCES[1] R. L. Burden and J. D. Faires,Numerical Analysis, 5th ed. Boston,MA: PWS-KENT, 1993.[2] E. Gelenbe, X. Mang, and R. Onvural, “Bandwidth allocation and calladmission control in high-speed networks,”IEEE Commun. Mag., pp.122–129, May 1997.[3] R. Guerin, H. Ahmadi, and M. Naghshineh, “Equivalent capacity andits application to bandwidth allocation in high-speed networks,”IEEEJ. Select. Areas Commun., vol. SAC-9, pp. 968–981, Sept. 1991.[4] J. S. Kaufman, “Blocking in a shared resource environment,”IEEETrans. Commun., vol. COM-29, pp. 1474–1481, Oct. 1981.[5] M. Ritter and P. Tran-Gia, “COST242 Interim report, multi-rate modelsfor dimensioning and performance evaluation of ATM networks,” Tech.Rep., Univ. Wurzburg, Germany, June 1994.[6] J. W. Roberts, “A service system with heterogeneous user requirement,”in Performance of Data Communications Systems and Their Applica-tions, G. Pujolle, Ed. Amsterdam, The Netherlands: North-Holland,1981, pp. 423–431.[7] K. W. Ross,Multiservice Loss Models for Broadband Telecommunica-tions Networks. London, U.K.: Springer-Verlag, 1995.[8] R. Syski, Introduction to Congestion Theory in Telephone Sys-tems. Edinburgh, U.K.: Oliver and Boyd, 1960.

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Performance analysis of single-link system with nonlinear equivalent capacity

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