The purpose of this paper is to supplement a recent paper by M. Rajaratnam and F. Takawira (see ibid., vol.49, p.817-34, 2000), which deals with a model for the performance analysis of cellular mobile networks. The key performance measure is a second-order characteristic (peakedness) of the traffic stream that serves as a model for handoff traffic. We show that this quantity may be obtained by evaluating an explicit formula rather than by solving a set of equations. This result enables us to verify some conjectures formulated by Rajaratnam and Takawira on the basis of numerical experiments. We also show the uniqueness of the solution to a system of nonlinear equations, required in the performance analysis, as conjectured by Rajaratnam and Takawira

Doorn, Erik A. van

Ta, An T.K.

English

van Doorn, Erik A.

Ta, A.T.K.

University of Twente Research Information

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 52, NO. 4, JULY 2003 953Proofs for Some Conjectures of Rajaratnam andTakawira on the Peakedness of Handoff TrafficErik A. van Doorn and An T. K. TaAbstract—The purpose of this paper is to supplement a recentpaper by Rajaratnam and Takawira, which deals with a model forthe performance analysis of cellular mobile networks. The key per-formance measure is a second-order characteristic (peakedness) ofthe traffic stream that serves as a model for handoff traffic. Weshow that this quantity may be obtained by evaluating an explicitformula rather than by solving a set of equations. This result en-ables us to verify some conjectures formulated by Rajaratnam andTakawira on the basis of numerical experiments. We also show theuniqueness of the solution to a system of nonlinear equations, re-quired in the performance analysis, as conjectured by Rajaratnamand Takawira.Index Terms—Carried traffic, cellular mobile network, handofftraffic, output process, output traffic, peakedness factor, smoothtraffic, tandem system.I. INTRODUCTIONI N a series of papers [1]–[5], Rajaratnam and Takawira haveproposed a method for analyzing the performance of cel-lular mobile networks. At the heart of their method, and morespecifically at the heart of their cell-traffic model, lies the simpletandem service system depicted in Fig. 1. It consists of a firstcell containing servers and a second cell with infinitely manyservers. Customers (calls) arrive at the first cell according to aPoisson process with intensity. If, upon arrival of a call, atleast one of the servers in the first cell is free, the call seizesan arbitrary free server and keeps it occupied during the call’sservice time; if all servers in the first cell are busy, the call islost. After being served in the first cell, a call leaves the systemwith probability 1 and is transferred with probabilityto thesecond cell, where it is served for a second time. Service timesare mutually independent random variables and also indepen-dent of the arrival process; the service times in cellare identi-cally distributed with mean , .The stream of calls entering the second cell models thehandoff traffic from a cell in a cellular network and is char-acterized by the mean and variance of thestationary number of occupied servers in the second cell. (Itwill be convenient to explicitly indicate dependence on.) Thequestion is how these quantities depend on the parameters ofthe model and, more concretely, whether explicit expressionsfor and can be given, perhaps in specific settings.Rajaratnam and Takawira have obtained partial solutionsto these problems. Also, they have formulated a number ofconjectures on the basis of numerical experiments. We will nowManuscript received September 4, 2002; revised March 24, 2003.The authors are with the Department of Applied Mathematics University ofTwente, P.O. Box 217, 7500 AE Enschede, The Netherlands (e-mail: e.a.van-doorn@utwente.nl; ttkieuan@yahoo.com).Digital Object Identifier 10.1109/TVT.2003.814216Fig. 1. A tandem service system.briefly describe their findings and the supplementary resultswe intend to establish in this paper.For future reference, we note that the meanand variance of the stationary number of occu-pied servers in the first cell (known as the mean and variance ofthecarried traffic) are given by(1)(2)where and denotes the Erlang lossfunction(3)(see, for example, [6]).It is shown in [4] and [5] that(4)(5)so that the problem of calculating and reduces tothat of finding and , the index“out” being mnemonic foroutput traffic. Evidently, by Little’sformula(6)but a relation between and has been established only inthe case that the service times in each cell are deterministic andequal, namely, (see [3]).Assuming that the service times are exponentially distributed,Rajaratnam and Takawira [4] show that may be obtainedby solving a set of 1 simultaneous equations. On the basisof numerical experiments, they subsequently conjecture that thepeakedness (factor) of the output trafficsatisfies(7)as or (8)0018-9545/03$17.00 © 2003 IEEE954 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 52, NO. 4, JULY 2003In Section II, we shall obtain an explicit expression for ,which will enable us in Section III to prove the conjecturedproperties of . It then follows immediately from (5) that thepeakedness factor of the handoff traffichas similar properties.As part of their (approximative) performance analysis, Ra-jaratnam and Takawira require a solution (, ) to the systemof equations and . Evidently, sucha solution will not exist in general if one requiresto be an in-teger. However, it is conceptually and computationally easy tocontinue the Erlang loss function to nonintegral valuesof the first argument by letting(9)(see, for example, [7] or [8]). With this interpretation of the Er-lang loss function, Rajaratnam and Takawira [4] conjecture thatthere is a unique solution (, ) to the system of equations(10)(11)for given values of and such that . In Sec-tion IV, we will verify this conjecture. (A similar problem in-volving overflow trafficwas solved in [8].)We conclude this introduction with the observation that theproof of (5) in [4] does not depend on the particular type oftraffic involved. That is, the effect on peakedness ofthinninganarbitrary traffic stream by accepting each call with probabilityis expressed by(12)where and denote the peakedness factors of the originaland the thinned stream, respectively. This relation was observedearlier in the context of exponential service times in [9].II. V ARIANCE OF THE OUTPUT TRAFFICSince service times are assumed to be exponentially dis-tributed, the number of calls in the first cell is a birth–deathprocess, and hence reversible (see, for example, Kelly [10]).It follows that, in equilibrium, the output process of the firstcell is stochastically identical to the input process, that is, thearrival process of served calls. (The input process together withthe overflow process—the arrival process of lost calls—con-stitute the arrival process to the first cell.) Calculating thevariance of the output traffic is therefore equivalent tocomputing the variance of the freed carried traffic,thatis, the variance of the stationary number of occupied serverswhen a copy of the input process to the first cell is offered toa hypothetical infinite-size cell in which each call is servedduring an exponentially distributed service time (of mean)that is independent of its service time in the first cell.Freed carried traffic has recently been studied in the moregeneral setting of renewal arrival processes by Brandt andBrandt [11]. Specifying their results for the setting at hand,these authors present in [11, (32)] an explicit formula for,which, in our notation, reads as(13)Here, as before, , is the Erlang lossfunction (3), and is the mean of the carried traffic givenin (1). In addition, is the mean of the freed carried traffic(the mean number of occupied servers in the hypothetical cell).Clearly, by Little’s law, , so thatin view of (6). Since , the solution to ourproblem of determining can now be given as follows.Theorem 1: The variance of the output traffic can berepresented as(14)where .We note that can be interpreted as the mean number of freeservers in the first cell, so that .For some specific parameter values, we have compared theresults of evaluating (14) with the results that Rajaratnam andTakawira [4, Fig. 7] obtained by solving a system of linear equa-tions, and found complete agreement.III. PROOFS FOR THEPEAKEDNESSCONJECTURESTheorem 1 tells us that the peakedness factorof the output traffic is given by(15)We can now prove the following theorem, verifying conjecture(8).Theorem 2: Let be a fixed positive integer; thenas or .Proof: It is obvious that if , then also, while .Hence the first part follows immediately from (15).To prove the second part, we note from (3) that, as ,we have(16)As a consequenceVAN DOORN AND TA: PROOFS ON THE PEAKEDNESS OF HANDOFF TRAFFIC 955so that(17)Evidently, if , then and , while, in viewof (16), . Since we can rewrite (15) asthese facts together with (17) imply that as ,as required.It requires more effort to prove the next theorem, which ver-ifies conjecture (7).Theorem 3: For any positive integer and real , wehave .Proof: We define(so that ) and let(18)Substitution of shows us that(19)Clearly, is a polynomial in of maximum degree 2 2, sothat we can write(20)We see from (15) and (18) that(21)and we will prove that by showing that each coefficientin (20) is nonnegative, while .We first note from (19) that , as required.Let us next assume that . It then follows from (19)thatThe following step is to observe from (19) thatWe subsequently obtain from (19) by straightforward calcu-lations thatso it remains to check whether for . Tothis end, we first derive a representation for , .Namely, from (19), we have956 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 52, NO. 4, JULY 2003so thatUsing this result twice, we obtain forSince , it now follows by induction that we havefor , . Thus we have shown that allcoefficients in (20) are nonnegative while . As a result,we have and hence, by (21), .We conclude this section with some remarks. First, the va-lidity of Theorem 3 was claimed earlier by Kirstein [12, (4.11)],but, as explained in a correction to [12] in [13], his proof is inerror. Then, recalling that the peakedness factorof the over-flow traffic satisfies(22)(see, for example, [6]), we note that (15) and Theorem 1 yieldthe interesting upper bound(23)supplementing the well-known lower bound (see, forexample, [8]). Conversely, the inequality readily leadsto the lower boundSince, by (1) and (2), the peakedness of the carried traffic isgiven by(24)this bound can nicely be expressed as(25)IV. UNIQUENESS OF ASOLUTIONAs announced, we will prove in this section the followingtheorem.Theorem 4: For given values of and such that, the system of (10) and (11) has a unique solution (, ) suchthat and .Proof: Writing and solving (11) for , wefindHence, lettingwe can reformulate our problem as that of establishing that thereis a unique positive solution to the single equationwhereIt is straightforward to verify that, under our assumption, we have for . Since asand as , the required result followsimmediately.We finally note that uniqueness of the solution was claimedalready by Katz in [14], but no proof was given. We must addthat Katz uses a logarithmic interpolation formula rather than (9)to compute the Erlang loss function for a nonintegral number ofservers.ACKNOWLEDGMENTThe authors thank B. Sanders of Vodafone Group Research &Development in Maastricht, The Netherlands, for pointing theirattention to the conjectures of Rajaratnam and Takawira. Theyare grateful to A. Brandt of Humboldt University, Berlin, Ger-many, for informing them about his joint work with M. Brandt[11] prior to publication.VAN DOORN AND TA: PROOFS ON THE PEAKEDNESS OF HANDOFF TRAFFIC 957REFERENCES[1] M. Rajaratnam and F. Takawira, “Hand-off traffic modeling in cellularnetworks,” in Proc. IEEE Globecom ’97, Phoenix, AZ, 1997, pp.131–137.[2] , “The two moment performance analysis of cellular mobile net-works with and without channel reservation,” inProc. IEEE ICUPC ’98,Florence, Italy, 1998, p. 1157.[3] , “Performance analysis of highway cellular networks using gener-alized arrival and generalized service time distributions,” inTeletrafficEngineering in a Competitive World (Proceedings of the 16th Interna-tional Teletraffic Congress), P. Key and D. Smith, Eds. Amsterdam,the Netherlands: Elsevier, 1999, pp. 11–22.[4] , “Nonclassical traffic modeling and performance analysis ofcellular mobile networks with and without channel reservation,”IEEETrans. Veh. Technol., vol. 49, pp. 817–834, 2000.[5] , “Handoff traffic characterization in cellular networks undernonclassical arrivals and service time distributions,”IEEE Trans. Veh.Technol., vol. 50, pp. 954–970, 2001.[6] A. Girard, Routing and Dimensioning in Circuit-Switched Net-works. Reading, MA: Addison-Wesley, 1990.[7] D. L. Jagerman, “Some properties of the Erlang loss function,”Bell Syst.Tech. J., vol. 53, pp. 525–551, 1974.[8] A. A. Jagers and E. A. van Doorn, “On the continued Erlang loss func-tion,” Oper. Res. Lett., vol. 5, pp. 43–46, 1986.[9] E. A. van Doorn, “Some aspects of the peakedness concept in teletraffictheory,” Elektron. Informationsverarb. Kybernet., vol. 22, pp. 93–104,1986.[10] F. P. Kelly, Reversibility and Stochastic Networks. Chichester, U.K.:Wiley, 1979.[11] A. Brandt and M. Brandt, “On the moments of overflow and freed carriedtraffic for the GI/M/C/0 system,”Methodol. Comput. Appl. Probab., vol.4, pp. 69–82, 2002.[12] B. M. Kirstein, “Some inequalities for queues with applicationsto tandem and overflow systems,”Elektron. Informationsverarb.Kybernet., vol. 14, pp. 445–462, 1978.[13] B. M. Kirstein, “Correction,”Elektron. Informationsverarb. Kybernet.,vol. 15, p. 336, 1979.[14] S. Katz, “Statistical performance analysis of a switched communicationsnetwork,” inProc. 5th Int. Teletraffic Congr., 1967, pp. 566–575.Erik A. van Doorn, received the M.Sc. degree inmathematics from Eindhoven University of Tech-nology, Eindhoven, The Netherlands, in 1974 andthe Ph.D. degree in mathematics from the Universityof Twente, Enschede,The Netherlands, in 1979.He was was with the Dr. Neher Laboratories,Netherlands Postal and TelecommunicationsServices, Leidschendam, from 1980 to 1982 andwith the Centre for Mathematics and ComputerScience, Amsterdam, the Netherlands, from 1982to 1985. Since 1985, he has been with the faculty ofmathematical sciences, University of Twente, as anAssociate Professor. His research interests range from fundamental issues inmathematical analysis and the theory of stochastic processes to performancemodeling and analysis of computer and communication systems.Dr. van Doorn serves on the editorial boards of theJournal of Communica-tions and Networks, Stochastic Models, andTelecommunication Systems.An T. K. Ta, was born in 1976 in Nghi Loc, Vietnam.She received the B.Sc. degree in mathematics fromVinh Pedagogic University, Vinh, Vietnam, in 1998.During the academic year 2000/2001, she par-ticipated in the master class program MathematicalFinance, offered by the Mathematical ResearchInstitute, The Netherlands. In 2002, she was withthe Department of Applied Mathematics, Universityof Twente, Enschede, The Netherlands, where sheworked on a project involving the modeling andperformance analysis of communication systems.Currently, she is with the University of Tilburg, Tilburg, The Netherlands,working in financial mathematics.

Proofs for some conjectures of Rajaratnam and Takawira on the peakedness of handoff traffic

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Hand-off traffic modeling in cellular networks,” in

Handoff traffic characterization in cellular networks under nonclassical arrivals and service time distributions,”

On the continued Erlang loss function,”

Performance analysis of highway cellular networks using generalized arrival and generalized service time distributions,”

Reversibility and Stochastic Networks.

Routing and Dimensioning in Circuit-Switched Networks.

S.Katz,“Statisticalperformanceanalysisofaswitchedcommunications network,” in

Some aspects of the peakedness concept in teletraffic theory,”

Some inequalities for queues with applications to tandem and overflow systems,”

The two moment performance analysis of cellular mobile networkswithandwithoutchannelreservation,”inProc.IEEEICUPC’98,

traffic modeling and performance analysis of cellular mobile networks with and without channel reservation,”

Proofs for some conjectures of Rajaratnam and Takawira on the peakedness of handoff traffic

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