softblock_vtc99

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Queueing model for soft-blocking CDMAsystemsLa c h la n L. H. Andr e w, Dona ld J. B. Payne and S te phe n V. Ha nly

De pa r tm e n t of Electr ical an d Ele c t r on ic Eng ine e r ingUniversity of Melb ourne , Parkvi l le, Vic 3052,AUSTRALIAPh+61393449208 Fax+61393449188{L.Andrew,djbp,S.Hanly}@ee.mu.oz.au

Abstract-A new model is proposed for the analy-sis of CDMA systems. Th is model is a birth and deathprocess whose birth process considers the new call ar-rival rate , the blocking rate, th e effect of soft hand offand th e effect of the hand allocation strategy in mu lti-band (multicarrier)CDMA systems. This model accu -rately predicts the distribution of the number of callsconnected to a base station.

I. INTRODUCTIONCode division multiple access (CDMA) systems

are characterised by soft blocking. New calls areblocked when the total same cell plus other cell in-terference is too high. T hus they can neither be con-sidered to he finite queues, which accept arrivals nn-til a fixed threshold is reached, or infinite queues,which never block arrivals. The traditional M/M/ccqueue model [ l] does not reflect this soft blockingbehaviour of a CDM A system. This paper developsan alternative mod el, in which each cell blocks newlyarriving calls with a state-dependent probability. Inits simplest form this model is similar to that used(but not directly evaluated) in [ 2 ] .

This mo del treats each cell of a multi-cell systemindependently. The state of the cell is the numberof users currently in that cell. Other cells simplycontribute random interference. This random inter-ference causes blocking with some probability, &,which is assumed to depend only on the state z of thecurrent cell.

The state of the cell can be modelled as a birthand death process [3]. Deaths (call departures) havenegative exponential inter-event times with rate pro-portional to the current number of calls. Births alsohave a negative exponential inter-event time, withrate equal to the arrival rate thinned by the blockingprobability, X ( l - Bi).In this paper, Bi are determined by simulation.

This work wa s funded in part by the Australian ResearchCouncil (ARC)

The question arises if Bi are from sim ulation, whymodel the system? Modelling each cell as indepen-dent, and treating all other cells as an independentrandom process is an approximation. If the simula-tion results agree with the Markov model, this vali-dates that approximation. In future, closed form ex-pressions for the Bis will he determined, and a fullyanalytic solution will h e available.

11. HARD H A N D O F FWhen hard handoff is used, new calls are allo-

cated to the nearest base station. This gives rise tothe very simple birth and death process with the ar-rival rate to state i being X ( l - Bi),nd the depar-ture rate ip, hown in Figure 1. Call this Markovchain Markov chain A. Figure 2 shows the proha-hility distribution for the state of a cell according toan event driven simulation of the system an d the statedistribution of Markov chain A. For comparison, italso shows th e distribution for an M/M/cc odel.Clearly the proposed model gives a much better fi tthan the traditional m odel. Th is simulation was for a4 x 4 hexagonal grid w ith a spreading factor of 128,and calls accepted if the signal to interference ratio(SIR) exceeds 6 dB. Log-normal shadowing was as-sumed, with standard deviation U = 8 dB, but mul-tipath fading was ignored. The load was 14Erlangs.The overall blocking probability was 12.7%, bothby simulation and the Markov A model. (Note thatvery high blocking probabilities are used through-out this paper to highlight the difference in the mod-els.) Whe n the blocking probability is low, it mayhe approximated by the probability that the numberof calls exceeds a maximum permissible value in anM/M/oo odel. The perm issible value typically usedis W /a- 1 - (1 B ) f X / p , where W is the pro-cessing gain, 01 is the required SIR, B is the over-all blocking probability, f is the ratio of same-cellto other-cell interference in a uniformly loaded sys-

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5 0.3 - -om 0.25 - -p 0.2 - -.-

.-Y0o 0.15 - -m 0.1 ..

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Fig. 2. State distribution for hard handoff: simulated,proposed Markov model and W o o odel.

tem, and A/p is the load on the system. Since thethreshold depends on the blocking probability, an it-erative solution is required (although in many casesthe approximation B N 0 suffices). For hard hand off,f Y 2.5, and in this case the blocking probab ility isapproximately 50%, the probability of exceeding ap-proximately 14 calls. Clearly the ap proximation thatthe distribution is approximately that of an M/M/ccqueue is not valid in this case, and thus the result issubstantially different from the true value and is ofno use in network planning.

Figure 3 shows the blocking probability in eachstate for the above simulation. As expected, theblocking probability is greater for heavily loadedcells than for cells with a mod erate load. A moreunexpected result is that blocking is also higher forvery lightly loaded cells. T his is because of the cor-relation in blocking events. If the other-cell interfer-ence i s high w hen a call arrives, it will probably alsobe high when the next call arrives. Thu s when theother-cell interference is high for a prolonged period,the occupancy of the cell will drop as users leavebut no new users are admitted. While it is surpris-ing that blocking is high when the num ber of calls ina cell is low, the converse (that the number of callsin a cell is low when the blocking is high) is intu-itively obvious. To verify this explan ation, the figurealso shows the blocking per state for soft handoff.(The load was increased to obtain per-state blockingof the same order of magnitude as for hard handoff,although the overall blocking was not matched.) In

soft handoff, the other cell interference is greatly re-duced since users are decoded by the base station atwhich they have the h ighest SIR. Hence the total in-terference will drop m ore markedly as the cell emp-ties than is the case for hard handoff, and prolongedperiods of high interference are less common. Thisphenomen on is not captured by the statistical modelsused in [ 2 ] and elsewhere.

111. SOFT A N D O F FIn soft handoff, a mobile connects simultaneously

to several base stations, and is decoded by the basestation with the highest SIR. This greatly improvessystem capacity. The state of a cell is then the num-ber of users which are currently being decoded bythat cell. Applying the blocking probabilities mea-sured fo r soft handoff to Markov chain A gives a verypoor fit to the soft handoff simulation. This can beseen in Figure 4, which shows the results for a load of20 Erlangs. The overall blocking rate is 7.3%, whichis the same order of mag nitude as the sim ulated valueof 5.6%. but that is largely due to the cancellationof the overestimate of blocking due to very heavilyloaded cells by the un derestimate of blocking due tomoderately loaded cells.

Soft handoff produces a distribution with morecells having approximately the average number ofcalls than hard h andoff. That is because calls willtypically have a higher SIR in more lightly loadedcells, and thus will be more likely to conn ect to thecorresponding base stations, leading to a higher ef-fective arrival rate at base stations with fewer calls.This can he mod elled by making the anival rate de-pend on the current state of the cell. A suitable andeffective heuristic for this amval rate will now he

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outlined.For this purpose it is reasonable to make the ap-proximation that all of the neighbouring cells have

an equal number of users n. Initially it will be as-sumed that n = A / p . It is also reasonable to assumethat the effective arrival rate will be proportional tothe true arrival rate, and so the effective amval ratein state i will be ki,,X( 1-Bi).Similarly, ki, , shouldnot chan ge when the entire load on the system scalesuniformly, so it will depend on i an d n only throughthe ratio i/n. When i /n 1, he system is uniformlyloaded, and the arrival rate at all cells will simply bethe true amva l rate, yielding IC , , , % 1. Fo r i / n >> 1,IC,,, will be almost zero, since most users will con-nect to neighbouring base stations, rather than thebase station of interest. Finally, for i /n 0. Em -pirically it has b een found that this provides a goodfit with a = 4. Figure 5 shows this scaling factoragainst i /n, he occupancy of the current cell rela-tive to the mean load. Thus the overall am val rateis

in state i . Call this modified M arkov chain Markovchain B . Once an estimate of the overall block-ing probability, B , is known, the Markov chain canbe improved by approximating the occupancy ofneighbouring cells as n N (X/k)(l - B) , yieldingMarkov chain B with two iterations. This mayh e repeated until the b locking probab ility converges.The results for soft handoff with a load of 20 Er-langs are shown in Figure 4. After one iteration,Markov chain B produced a much better approxima-tion to the true state distribution than M arkov chainA. How ever, it slightly overestimated the number ofcalls in the cell (that is, the mean of the distributionwas shifted to the right), and it predicted an overallblocking rate of 7.4%, which is very similar to thatof Markov ch ain A, but substantially different fromthe true value of 5.6% obtained by simulation. Aftera second iteration, the model produced a very goodapproximation to the simulated state distribution, andpredicted a blocking probability of 5.4%. For com-parison, the results with no blocking are included.

The fact that results from Markov chain B matchthe simulated curve well but those from Markovchain A do not show s that the good m atch is not in-

Markov B0.12 1no blocking - !/ \>i i0.1 t

0 5 10 15 20 25 30# Calls in cellFig. 4. State distribution for soft handoff: Simulated,original Markov chain, mod ified Markov chain.

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0 0.5 1 1.5 2 2.5 3 3.5 4relative lo ad, i/nFig. 5 . Scaling factor applied to arrival rate, as a function

of i ln .

evitable given the per-state blocking probabilities,but rather that the choice of Markov chain is respon-sible for the good m atch.

Note that it was not necessary to alter the anivalrates in this way for soft handoff in [2].That is be-cause base station selection there was based on themeasured path gain, rather than the SIR. Thus thebase station to which a new call connected depend edonly on the propagation conditions and was indepen-dent of the load on the cell, causing the arrival rate(before blocking) to be indpendent of the state.

Iv. M U L T I - B A N DPERATIONThe capacity of a CDM A system may be easily in-creased using multi-band (or multi-channel) CDMA ,in which a secon d slice of spectrum is used, and callsare allocated to one or other band [MI. he perfor-

mance of su ch systems is influen ced by the way newcalls are allocated to bands, and this difference inperformance can be predicted by th e proposed tech-nique. Th e Markov chain will now be two dimen-

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P : 28 : I N :Fig. 6. Markov chain for tw o band system.

sional (or N dimensional for N bands). For simplic-ity, this pape r will only treat the case of hard handoff.One obv ious strategy is to assig n a call to a band

randomly. If the call is blocked in the chosen band, itis tried in the n ext band. This gives rise to the M arkovchain of Figure 6, where X i j is the arrival rate to theband in state j rom either state ( i , j )or (j ,i) . Therate of arrivals to band k from a given state is theoverall arrival rate (= A) times the probab ility of ini-tially selecting band k (= 1/2), times the probabilityof not being blocked on band k given that band k is instate j (= 1- Bj),plus the overall arrival rate timesthe probability of initially selecting the other band(= 1/2), but being blocked on that band (= Bi ) an dnot being blocked on band k (= 1-Bj) .Thus X i j =By symmetry, this collapses to the chain of Figure 7,but now with Xi; doubled to X ( l - B:). That is be-cause here the probability of being in state ( i , ) isthe probability of having i calls in each of the twobands, while the probability of being in state ( i , j ) ,j # i , is the probability of either having i calls inthe first hand an d j n the second, or having j calls inthe first band and i in the second. N ote that this doesnot require knowledge of the blocking probability foreach state, but only the ma rginal blocking probabilitygiven there are i calls in a particular band.

Another strategy which has proven successful isto assign new calls to the band which currently hasthe fewest calls [ 5 ] , with blocked calls again tryingthe other band. Th is can again be represented by thecollapsed Markov chain of Figure 7, but now withX i j = X ( 1 - Bj) if i > j , X i j = X B i ( 1 - Bj)

(x /2) ( (1-B, )+Bi( l -B , ) ) = X ( l - B j ) ( l + B J / 2 .

Fig. 7. CollapsedMarkov chain for tw o band system

if i < j , an d X i i = X ( l - B:). This least load(LC) strategy has been analysed thoroug hly underthe assumption of no blocking [6].

Figures 8 an d 9 show the actual (i.e., simulated)and Markov state distributions for each of the twostrategies, along w ith the M/M/cc results for compar-ison. Th e spreading factor was halved to 64, so thatthe same total bandwidth was used as fo r the singleband case, and the offered load was 17 Erlangs. Forsimplicity, the graph show s the marginal probabilityof having i calls in the first band, rather than the fulldistribution. Clearly there is very good agreementbetween the model and simulation. Least load allo-cation show s a mo re peaked distribution, while ran-dom allocation sh ows a heav ier tail, with mo re cellshaving a large number of calls. Th is behaviour iscaptured well by the proposed Markov m odel.

The blocking probabilities for the simulated (re-spectively Markov) cases were 16.1% (15.6%) forleast load, and 16.6% (13.4%) for random. The fig-ures for the Markov model are an approximation,given by the square of the marginal per-band block-ing probability. This i s a fair approximation becausea call must be rejected on both band s to be blocked.

To show once again that the good agreem ent withsimulation is not solely due to the use of accurateper-state blocking probabilities, Figures 10 and 11show the Markov distribution for each strategy us-ing the per-state blocking probabilities from the otherstrategy. E ach graph show s the two sets of simula-tion results and the results using on e of the Markovchains with both sets of measured blocking probabil-ities. In each case, the predicted results match the

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0 2 4 6 8 10 12 14 1 6 18Number of CallsFig. 8. Simulated vs. Markov and M/M/m results forrandom band allocation strategy.

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-~ .. ,0.020 . , , , , , 14 16 1810 12O .,.. L...zc .,,., V , I 1 ~ I I wallsFig. 9. Simulated vs. Markov and M/M/m results forleast load band allocation strategy.

simulation to which the Markov model correspondsmore closely than the one from which the per-stateblocking probabilities were taken. This confirms thatthe model is not overly sensitive to the exact per-stateblocking probabilities. Thus this mod el can be usedto predict the overall blocking probability for differ-ent band allocation schemes based on the differentMarkov chains and a single set of measured per-stateblocking probabilities.

V. CONCLUSIONIt is possible to model soft blocking in m ulti-cellCDMA systems as an independent birth and death

process at each cell. The birth process is thinned bythe state-dependen t blocking probability. Th e birthprocess can also be modified to model soft handoff,and the process can be extended to analyse multi-band systems.

0 .18 , I I I , , I I , I I

Fig. 10. Predicted state distributionsfor random Markovmodel using least load blocking probabilities.

Sim, RndSim, LL . - - - -.160.14 :_,--._:: -\ ,,

0 2 4 6 8 10 12 14 16 18Numberof CallsFig. 11. Predicted state distributions for least loadMarkov model using random blocking probabilities.

AC KNOWLEDGEM ENTTh e authors thank Chaitanya Ra o for commen ts on

this manuscript, and for pointing out an error in theoriginal arrival rates for the mu ltiband Markov chainwith random assignment.

REFERENCES[ I ] A. I. Viterbi, Principles of Spread Spectrum Communica-tion. Reading, M A: Addison-Wesley, 1995.[2] Y Ishikawa and N. meda, Capacity design and perfor-mance of call admission control in cellular CDMA systems,

IEEE J. Select. Areas Comniun., vol. 15, pp. 1627-1635,Oct. 1997.

[3] M. IOjima, Markov proces ses for stochastic modeling. Lon-don: Chapman and Hal, 1997.

141 L. L. H. Andrew, Measurement-based band allocation inmultiband CDMA, in Proc. Infocom 99, (New York),1999,pp. 1536-1543.

[SI T. Dean, P. Fleming, and A. Stolyar, Estimates of multi-carrier CDMA system capacity, n Proc. Winter Sim. Conf,(Washington,DC), 1998,pp . 1615-1622.[61 P. J. Fleming an d B. Simon, Heavy traffic approximationsfor a system of infinite servers with load balancing, to ap-

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