A2.5 A Simplified Blocking Probability Calculation in the Retry Loss ...

A Simplified Blocking Probability Calculation inthe Retry Loss Models for Finite SourcesIoannis D. Moscholios, Michael D. Logothetis and George K. KokkinakisWCL, Dept. of Electrical & Computer Engineering, University of Patras, 265 00 Patras, GreeceE-mail: {moscholios, m-logo, gkokkin}@wcl.ee.upatras.grAbstract-We consider the single and multi-retry loss modelsfor finite sources (f-SRM, f-MRM, respectively) in whichcalls of K service-classes are accommodated in a link ofcapacity C and compete for the available bandwidth underthe complete sharing (CS) policy. Blocked calls canimmediately retry one or more times to be connected in thesystem with reduced bandwidth and increased service timerequirements. In both f-SRM and f-MRM the calculation ofthe link occupancy distribution can be quite complex since itrequires enumeration and processing of the state space. Tosimplify this calculation and consequently the calculation ofblocking probabilities, we present an approximate methodbased on the corresponding retry loss models of infinitesources. Simulation results verify the method’s accuracy.Keywords: Loss System; Finite Population; Retrials; BlockingProbability; Recurrent Formula.I. INTRODUCTIONThe wireless multi-service networking environmentnecessitates the use of teletraffic models with finite sourcepopulation for the assessment of the call-level QoS [1].The basic multi-rate loss model for finite sources refersto fixed-rate traffic and we name it Engset Multirate LossModel (EnMLM), since for a single service-class, it givesthe same blocking probability results with the Engsetformula for the time congestion probability [2]. Accordingto the EnMLM a link of capacity C accommodates calls ofK different service-classes under the CS policy, i.e. callsare accepted in the system if their required bandwidth isavailable. The offered traffic load of each service-class k(k=1,…,K) comes from a finite number of N k sources. Themodel has a product form solution (PFS) and thecalculation of the link occupancy distribution G(j), where jis the occupied link bandwidth, is based on an accurateand recursive formula [3].Extensions of the EnMLM which cope with elastictraffic are: a) the f-SRM [4] and the f-MRM [5], in whichblocked calls can immediately retry once or many times tobe connected in the system with reduced bandwidth andincreased service time requirements and b) the thresholdmodels where calls, prior to being blocked, may adjusttheir bandwidth and service time requirements accordingto either a single threshold, or a set of thresholds commonto all service-classes or a different set of thresholds foreach service-class [5]. An important consideration formulti-service networks is the application of the bandwidthreservation policy in the above mentioned models [1],[6].In all these extensions the PFS is destroyed and thereforeonly approximate but recursive G(j)’s formulas areavailable. Furthermore, in both the EnMLM and itsextensions the G(j)’s determination is more complexcompared to that in the corresponding infinite sourcemodels ([7]-[10]), since the system’s state space needsenumeration and processing prior to the G(j)’s calculation.In the case of the EnMLM one can overcome the statespace enumeration and processing by an approximatemethod proposed, in [11], by Glabowski and Stasiak(G&S method) which is based on the correspondinginfinite source model, the classical Erlang Multirate LossModel (EMLM) [12]. According to [11], the number ofin-service sources, n k , of service-class k in a state j of anEnMLM system is approximated by the average number,y k (j), of service-class k calls in the same state j of thecorresponding EMLM system. The values of y k (j) areeasily computed via the well-known Kaufman-Robertsformula ([7],[8]) and then replace the corresponding n k ’sin the G(j)’s calculation of the EnMLM.In this paper we study the applicability of the G&Smethod in the case of the f-SRM and f-MRM. Itsapplication requires the knowledge of the average numberof service-class k calls in each state j from thecorresponding single and multi-retry models (SRM andMRM, respectively) for infinite number of sources [9].The analytical blocking probability results obtained by theG&S method are in accordance with simulation results.The remainder of the paper is as follows: In Section IIwe review the f-SRM and the f-MRM. In Section II.A wepresent the analytical model of the f-SRM while inSection II.B the analytical model of the f-MRM. InSection II.C we present, through a simple example, themethod for the state space determination in the case of thef-SRM. In Section III we proceed with the application ofthe G&S method in both models. Numerical results arepresented in Section IV. In Section V we conclude.II. THE RETRY MODELS FOR FINITE SOURCESA. The f-SRM - the analytical modelWe assume a link of capacity C bandwidth units (b.u.)that accommodates calls of K service-classes whichcompete for the available bandwidth under the CS policy.Each service-class k has a finite source population N k andrequires b k b.u per call. Service-class k calls arrive to thelink according to a quasi-random process [2], with meanarrival rate λ k =(N k - n k )v k , where n k is the number of inservicesources and v k is the mean arrival rate per idlesource. The offered traffic load per idle service-class k−1source is α k =v k /µ k where µ k is the mean service time,exponentially distributed. Blocked service-class k callsretry to be connected in the system with parameters (b kr ,µ ) where b kr < b k and µ > µ . The f-SRM does not−1kr−1krhave a PFS and therefore the G(j)’s calculation is based onan approximate recursive formula [4]:−1k

⎧1 for j = 0⎪K1⎪ ∑(N−n+ 1)αk bkG(j -bk) (1)⎪j k kk= 1G(j)= ⎨K⎪ 1+⎪ ∑(N−(n+ n ) + 1)αkrbkrγk(j)G(j-bkr) for j = 1,...,Cj k k krk= 1⎪⎩0 otherwisewhere: n kr is the in-service retry calls of service-class k,−1α kr =v kr µ kr (offered traffic load per idle source of retryservice-class k calls), γ k (j)=1 when j >C-(b k -b kr ).The proof of eq.(1) is based on two assumptions [4]: 1)the existence of local balance, which appears only in PFSmodels and 2) the application of the migrationapproximation (M.A.) which assumes that the populationof retry service-class k calls are negligible when j ≤C-(b k -b kr ). The existence of M.A. in eq.(1) is expressed by thevariable γ k (j). Call blocking probabilities (CBP) ofservice-class k, B kr , is the probability of a call to beblocked with its retry bandwidth and is given by:C−1CB = ∑ G G(j) , wherekrG = ∑ G(j)(2)j= C− bkr+ 1j=0Note: By the term CBP we mean time congestionprobabilities. For quasi-random arrivals, there is adistinction between time and call congestion probabilities.These probabilities coincide in the case of Poisson arrivals(PASTA property [2]). Time congestion probability isdetermined by the proportion of time the system iscongested and can be measured by an outside observer.When N k approaches infinity for k =1,…,K then theSRM results. In that case the G(j)’s calculation is based onthe following approximate but recursive formula [9]:⎧1 for j = 0⎪K⎪ 1⎪ ∑α k b k G(j - b k )jk = 1(3)G(j) = ⎨K⎪ 1⎪+γ (j)G(j - ) for j = ,..., C∑ α kr b kr k b kr 1⎪ jk = 1⎪⎩0 otherwise−1−1where: α k =λ k µ k ,α kr =λ k µ kr and γ k (j)=1 when j>C-(b k -b kr ).The proof of eq. (3) is based on the assumptions of eq.(1), while the CBP of service-class k, B kr is determined byeq. (2). Based on eq. (3), one can calculate, in state j, theaverage number of service-class k calls either with b k b.u.,y k (j), or with b kr b.u, y kr (j):y ( j)= a G(j − b ) G(j)(4)kkwhere: j-b k > 0kykr( j)= akrγ k ( j)G(j − bkr) G(j)(5)where: j-b kr > 0 and γ k (j)=1 when j >C-(b k -b kr ).B. The f-MRM - the analytical modelIn the f-MRM, a blocked service-class k call may retrymany times with “retry parameters” (b krl , µ -1 krl ) for−1l=1,…,s(k), where b krs(k) −1µ k .µ krs(k )Similar to the f-SRM, the f-MRM does not have a PFSand therefore the G(j)’s calculation is based on thefollowing approximate recursive formula [5]:⎧1for j = 0⎪ K⎪1⎪ ∑[Nk−nk+ 1] akbG k ( j−bk)⎪jk=1G(j)= ⎨K s(k)⎪ 1⎪+∑∑[ Nk−(nk+ nkr+... + nkr+... + nl⎪ j1k= 1 l=1⎪⎩0otherwisekrs( k)) + 1] a b γ ( j)G(j−b) forj = 1,..., Ckrlkrlkrlwhere: n krl is the in-service retry service-class k calls withb krl , α krl =v krl µ -1 krl , while γ kl (j)=1 when C ≥ j>C-(b krl-1 - b krl ).The CBP of a service-class k, denoted as B krs(k) , is theprobability of a call to be blocked with its last bandwidthrequirement and is calculated by:Bkrs(k)krl(6)C1∑ G−C= G(j), where G = ∑ G(j) (7)j=C−bkr s+ 1( k )j=0When N k approaches infinity for k =1,…,K then the MRMresults and the G(j)’s calculation is based on [9]:⎧1for j = 0⎪ K⎪ 1⎪ ∑a k bkG(j − bk)⎪jk = 1G(j)= ⎨K s(k )⎪ 1⎪+∑∑akr blkr γl⎪ jk = 1 l = 1⎪⎩0otherwisek l( j)G(j − bkrl) for j = 1,..., C−1where: α k =λ k µ k, α krl =λ k µ -1 krl and γ kl (j)=1 when C≥j>C-(b krl-1 -b krl ), otherwise γ kl (j) = 0.Having determined the values of G(j)’s through eq. (8),then CBP of service-class k calls, B krs(k) , are given by eq.(7), the average number of service-class k calls, in state j,with b k b.u., y k (j), by eq. (4) and the average number ofservice-class k calls with b krl b.u, y kr l(j) by the equation:ykr( j)= a ( j)G(j b ) G(j)l kr γl k −lkr(9)lwhere: j-b krl >0 and γ kl (j)=1 when j>C-(b krl-1 –b krl ),otherwise γ kl (j) = 0.C. State space determination and G(j)’s calculation in thef-SRMConsider a link of capacity C = 5 b.u. and two serviceclasseswhose calls require b 1 =3 and b 2 =2 b.u.,respectively. The offered traffic load per idle source is α 1= α 2 = 0.01 erl, while the number of sources is N 1 =N 2 =6.Blocked 1 st service-class calls retry with reducedbandwidth, b 1r =1 b.u, and increased offered trafficα 1r =0.03 erl, so that the total offered traffic load per idlesource remains the same (α 1 b 1 = α 1r b 1r ). Due to the M.A.retry calls of the 1 st service-class are assumed to benegligible when the occupied link bandwidth j≤C–(b 1 –b 1r )⇒ j ≤ 3. Taking into account the M.A. assumption, thestate space consists of 8 states (n 1 , n 2 , n 1r ), presented in(8)

Table I together with the respective occupied linkbandwidth (j) and the blocking states (B 1 , B 2 , B 1r ), whereB k (k=1,2) expresses the CBP of service-class k calls withb k b.u, calculated by eq. (2) if one replaces b kr with b k .According to Table I the values of j = 4, 5 appear morethan once, and therefore it is impossible to use directly eq.(1) for the calculation of G(j)’s (e.g. the value j = 4,corresponds to both (n 1 , n 2 , n 1r ) = (0,2,0) and (1,0,1)). Toovercome this problem, the authors proposed a methodwhereby an equivalent stochastic system is determinedwith the following three characteristics [5]:a) The states (n 1 , n 2 , n 1r ) of the equivalent system are thesame with the initial one.b) Each state (n 1 , n 2 , n 1r ) of the equivalent system has aunique value for the occupied link bandwidth.c) The chosen values of b 1 , b 2 and b 1r of the equivalentsystem keep constant (as much as possible) the initialratios of b 1 :b 2 :b 1r :C = 3:2:1:5.In our example, the values b 1 = 3000, b 2 = 2000, b 1r =1001 and C= 5002, is an approximate solution to theinitial system; for these values we present in the lastcolumn of Table I the unique values of the equivalentoccupied link bandwidth, j eq . The resultant CBP are: B 1 =5.98%, B 2 = 0.66% and B 1r = 0.33%. Note that everysystem that is a multiple of b 1 = 3000, b 2 = 2000, b 1r =1001 and C= 5002 (e.g. b 1 = 6000, b 2 = 4000, b 1r = 2002and C= 10004) gives exactly the same CBP.The determination of an equivalent system, and thusCBP, can become difficult when calls retry many times.To circumvent this problem we examine, in the nextsections, the applicability of the G&S method in the f-SRM and f-MRM.TABLE ISTATE SPACE (n 1 , n 2 , n 1r ), OCCUPIED LINK BANDWIDTH j, BLOCKINGSTATES B 1 , B 2 , B 1r AND EQUIVALENT OCCUPIED LINK BANDWIDTH j eq .n 1 n 2 n 1r j B 1 B 2 B 1r j eq0 0 0 0 00 1 0 2 20000 2 0 4 * * 40000 2 1 5 * * * 50011 0 0 3 * 30001 0 1 4 * * 40011 0 2 5 * * * 50021 1 0 5 * * * 5000III. APPLICATION OF THE G&S METHOD IN THE RETRYMODELS FOR FINITE SOURCESAccording to the G&S method, initially proposed in thecase of the EnMLM [11], one approximates the number ofin-service sources, n k , of service-class k in a state j of anEnMLM system with the average number, y k (j), ofservice-class k calls with b k b.u. in the same state j of anEMLM system.Similar to the above, we consider again the simple f-SRM example of the previous section and approximate,for each state j = 1,…,5, the number of in-service sourcesn 1 , n 2 , n 1r with the average number y 1 (j), y 2 (j), y 1r (j),respectively, of the corresponding SRM system. Note thatin the SRM system: α 1 = α 2 = 0.06 erl and α 1r =0.18 erl.The values of y 1 (j), y 2 (j) are given by eq. (4) while thoseof y 1r (j) by eq. (5). The results are summarized in Table II.TABLE IIAVERAGE NUMBER OF CALLS IN EACH STATE j IN THE SRM SYSTEMy 1 (j) y 2 (j) y 1r (j) j0 0 0 00 0 0 10 1 0 21 0 0 30 0.8 2.4 40.9569 0.9569 0.2153 5Having calculated y( )’s we continue by determining thelink occupancy distribution G(j) of the f-SRM example,based on the following equation:⎧⎪⎪⎪G(j) = ⎨⎪ 1⎪+⎪ j⎪⎩K∑k= 11j∑(N −y (j-b)) α b G(j -b )(N −(y (j-b) + y (j-b))) α b γ (j)G(j -b ) forj= 1,...,CkkkrKk= 1kr1 forj = 00kkrkkkkr kr kotherwisekkkr(10)−1where: α kr =v kr µ kr , γ k (j)=1 when j >C-(b k -b kr ).The resultant CBP are: B 1 = 5.98%, B 2 = 0.66% and B 1r =0.33%, which are almost identical to those obtained by theequivalent stochastic system. Note that the correspondingsimulation results for the f-SRM example are: B 1,sim =5.79%, B 2,sim = 0.65% and B 1r,sim = 0.33%.In the case of the f-MRM the corresponding (to eq.(10)) equation for the G(j)’s calculation is the following:⎧⎪⎪⎪G(j)= ⎨⎪ 1⎪+⎪ j⎪⎩K s(k)∑∑k= 1 l= 11j∑(N −Y (j-bkKk= 1kfor j =(N − y (j-b)) α b G(j -b )kkrlkk)) α b010kkrlkrlklγ (j)G(j-botherwisekkkrl) for j= 1,...,C(11)where: α krl =v krl µ krl -1 , γ kl (j)=1 when C ≥ j>C-(b krl-1 -b krl ) andY ( j− b ) = y ( j−b) + y ( j−b) + ... + y ( j−b) + ... + y ( j−b).kkrlkkrlkr1 krlkrlkrlkrs( k)krlIV. NUMERICAL EXAMPLEIn this section we compare the analytical CBP resultsobtained by the application of the G&S method in an f-MRM example with simulation results, which are meanvalues of 7 runs with 95% confidence interval.Consider a link of capacity C=50 b.u. and two serviceclasseswhich require b 1 = 10 and b 2 = 7 b.u., respectively.The offered traffic load (per idle source) of each serviceclassis α 1 =0.06 erl and α 2 =0.2 erl, respectively. Blockedcalls of the 1 st service-class reduce their bandwidthrequirement two times, from 10 to 8 and to 6 b.u. In thefirst case the offered traffic load is α 1r1 =α 1 b 1 /b 1r1 =0.075 erlwhile in the second α 1r2 =α 1 b 1 /b 1r2 =0.1 erl. Blocked calls ofthe 2 nd service-class reduce their bandwidth requirementonce, from 7 to 4 b.u. The retry offered traffic load of

second service-class calls is α 2r1 =α 2 b 2 /b 2r1 =0.35 erl. Thenumber of sources for both service-classes is N 1 = N 2 =12.The equivalent stochastic system used for the CBPcalculation is the following [5]: C=50006, b 1 =10000,b 2 =7001 b 1r1 =8000, b 1r2 =6000 and b 2r1 =4000. Table IIIshows the state space which consists of 30 states.The corresponding MRM system used for the y( )’s andCBP calculation, according to the G&S method, is:C=50 b.u., b 1 =10 b.u., α 1 =0.72 erl, b 1r1 =8 b.u., α 1r1 =0.9erl, b 1r2 =6 b.u., α 1r2 =1.2 erl and b 2 =7 b.u., α 2 =2.4 erl,b 2r1 =4 b.u., α 2r1 =4.2 erl.In Table IV, we present the analytical CBP resultsobtained from the equivalent stochastic system and theG&S method together with the corresponding simulationresults. At each point (P) of Table IV (P= 1,…,6), thevalues of α 1 , α 1r1 , α 1r2 are constant (i.e. α 1 =0.06,α 1r1 =0.075 and α 1r2 =0.1 erl) while those of α 2 , α 2r1 areincreased by 0.4/12 and 0.7/12 respectively, (for P=1:α 2 =0.2 and α 2r1 =0.35 erl).According to the results of Table IV, one observes thatthe G&S method gives almost the same results with theequivalent stochastic system’s method. This behaviour hasbeen observed in many examples. Further study in thecase of more complicated examples or in the case of thethreshold models for finite sources is needed in order toverify the superiority of the G&S method.V. CONCLUSIONWe review the f-SRM and the f-MRM loss models whereblocked calls try to be connected in the system withreduced bandwidth and increased service-timerequirements. The calculation of the link occupancydistribution in these models and therefore of CBP is basedon the enumeration and processing of the state spacewhich can be complex in systems where the number ofretrials is large. To simplify the CBP calculation weexamine the applicability of the G&S method, alreadyproposed in the case of the EnMLM, in both models.Results show that the G&S performs quite well.REFERENCES[1] I. Moscholios, M. Logothetis and M. Koukias, “A State-DependentMulti-Rate Loss Model of Finite Sources with QoS Guarantee forWireless Networks”, to appear in the Mediterranean Journal ofComputers and Networks, 2006.[2] H. Akimaru, K. Kawashima, “Teletraffic – Theory andApplications”, 2 nd edition, Springer, Berlin, 1999.[3] G. Stamatelos, J. Hayes, “Admission control techniques withapplication to broadband networks”, Computer Communication,vol. 17 , no. 9, September 1994, pp. 663-673.[4] G. Stamatelos, V. Koukoulidis, “Reservation-based bandwidthallocation in a radio ATM network”, IEEE/ACM Trans. Networkingvol. 5, no.3, June 1997, pp. 420-428.[5] I. Moscholios, M. Logothetis and P. Nikolaropoulos, “Engset multiratestate-dependent loss models”, Performance Evaluation, vol. 59,issues 2-3, February 2005, pp. 247-277.[6] I. Moscholios, M. Logothetis, “Engset multi-rate state-dependentloss models with Qos guarantee”, International Journal ofCommunications Systems, Vol. 19, Issue 1, February 2006, pp. 67-93.[7] J.S. Kaufman, “Blocking in a shared resource environment”, IEEETransactions on Communications, vol.29, no. 10, October 1981,pp.1474-1481.[8] J.W. Roberts, “A service system with heterogeneous userrequirements”, in: G. Pujolle (Ed.), Performance of DataCommunications systems and their applications, North Holland,Amsterdam, 1981, pp. 423-431.[9] J.S. Kaufman, “Blocking with retrials in a completely sharedresource environment”, Performance Evaluation, vol.15, issue 2,June 1992, pp. 99-113.[10] I. Moscholios, M. Logothetis, G. Kokkinakis “ConnectionDependent Threshold Model: A Generalization of the ErlangMultiple Rate Loss Model”, Performance Evaluation, vol. 48, issue1-4, pp. 177-200, May 2002.[11] M. Glabowski, M. Stasiak, “An Approximate Model of the Full-Availability Group with Multi-Rate Traffic and a Finite SourcePopulation”, 12 th GI/ITG Conference on Measuring, Modelling andEvaluation of Computer and Comummunication Systems (MMB)and 3 rd Polish-German Teletraffic Symposium (PGTS),MMB&PGTS 2004, Dresden, Germany, 12-15 September 2004.[12] K.W. Ross, “Multiservice Loss Models for BroadbandTelecommunications Networks”, Springer, London, 1995.TABLE IIISTATE SPACE (n 1 , n 2 , n 1r1 , n 1r2 , n 2r1 ), OCCUPIED LINK BANDWIDTH j,BLOCKING STATES B 1r2 , B 2r1 AND EQUIVALENT OCCUPIED LINKBANDWIDTH j eq .n 1 n 2n 1r1 n 1r2 n 2r1 j B 1r2 B 2r1 j eq0 0 0 0 0 0 00 1 0 0 0 7 70010 2 0 0 0 14 140020 3 0 0 0 21 210030 4 0 0 0 28 280040 5 0 0 0 35 350050 6 0 0 0 42 420060 6 1 0 0 50 * * 500060 7 0 0 0 49 * * 490071 0 0 0 0 10 100001 1 0 0 0 17 170011 2 0 0 0 24 240021 3 0 0 0 31 310031 4 0 0 0 38 380041 5 0 0 0 45 * 450051 5 0 0 1 49 * * 490052 0 0 0 0 20 200002 1 0 0 0 27 270012 2 0 0 0 34 340022 3 0 0 0 41 410032 3 1 0 0 49 * * 490032 4 0 0 0 48 * * 480043 0 0 0 0 30 300003 1 0 0 0 37 370013 2 0 0 0 44 440023 2 0 0 1 48 * * 480023 2 0 1 0 50 * * 500024 0 0 0 0 40 400004 1 0 0 0 47 * * 470015 0 0 0 0 50 * * 50000TABLE IVANALYTICAL AND SIMULATION CBP RESULTSEquivalent G&SSimulationP Stochastic System methodB 1r2 (%) B 2r1 (%) B 1r2 (%) B 2r1 (%) B 1r2 (%) B 2r1 (%)1 3.05 2.03 3.05 2.03 3.23±0.49 1.83±0.132 4.32 2.72 4.32 2.72 4.02±0.32 2.44±0.123 5.78 3.50 5.78 3.50 5.19±0.32 3.20±0.244 7.39 4.36 7.39 4.36 7.01±0.30 3.76±0.295 9.12 5.30 9.13 5.30 8.16±0.49 4.71±0.356 10.94 6.31 10.94 6.31 9.58±0.44 5.67±0.22