Velocity-assisted paging in PCS through stochasticmodelling of terminal displacement

P.G. Balis and O.R. Hinton

Abstract: The authors propose a novel framework for the exploitation of mobile velocity for theoptimisation of paging performance in PCS. The focus is on the optimisation of the paging processbased on the estimation of terminal displacement from its last known position. This estimation isperformed through a stochastic formulation, making use of both the velocity category of theterminal and the velocity category residence time. Simulation results are provided and the proposedmethod is compared with other paging strategies, indicating certain advantages that together withits very low complexity could make it applicable for both 2G and 3G systems.

1 Introduction

The issue of mobility management for cellular communica-tions systems has received considerable attention during thepast few years. In particular, the optimisation of pagingperformance and paging cost have been treated in severalpublished papers, e.g. [19], providing a framework for theefcient handling of the rapidly increasing signalling loadrelated to mobility management functions in modernmobile communications systems. In this context, variouspaging schemes have been proposed and their performancehas been investigated under certain assumptions regardingthe user mobility and trafc prole.Classifying the available paging strategies according to

the relevant overview presented in [10], blanket paging is thesimplest form of paging in which all the cells within thelocation area (LA) in which the mobile station (MS) islocated are paged simultaneously when a call arrives. Thisstrategy yields the lowest value of paging delay since theterminal is located within a single paging cycle, with thedrawback of high base transceiver station (BTS) paging costsince all cells in the LA are paged. The sequential pagingscheme is a strategy in which the LA is split into pagingareas (PA) and in the case of a mobile terminated call a PAis rst paged. If the called party is not located in oneattempt, paging is carried out sequentially in all PAs, untilthe locating process is terminated. The sequential pagingstrategy will provide decreased BTS paging cost at a cost ofincreased paging delay. An evolution of the sequentialpaging scheme is the sequential paging based on userslocation probability scheme, in which the current locationof the mobile terminal is predicted based on its locationprobability distribution. The BTS paging cost is thenminimised by sequentially polling the PAs in decreasingorder of probability. The main open issue with this strategyis the lack of a simple framework for the estimation of thetime-varying user location probability. Such an estimationcould be considerably assisted by the availability of

information regarding the mobile velocity, giving rise tovelocity-assisted paging strategies. Such paging schemeshave been presented in [8] and [1]. In both these reportspaging optimisation is performed through knowledge of thevelocity category of the user, with three and two mobilevelocity categories, respectively. The mobile velocity cate-gory is updated at the last interaction of the terminal withthe network and is assumed constant up to the pagingattempt. However, the assumption of constant mobilevelocity category may severely deviate from the real userbehaviour, particularly for large periods of no interactionwith the network where the user could change mobilevelocity category several times. On the other hand, thisassumption does not allow a realistic investigation of thedependence of the performance of intelligent pagingstrategies on the time elapsed from the last interaction ofthe terminal with the network.In this work we propose a novel approach for the

exploitation of the knowledge of terminal velocity for theoptimisation of paging performance. Apart from theknowledge of the mobile velocity category we consider asessential input the time during which the terminal remainsin each mobility category. Based on the above input, wepropose a stochastic framework for the estimation ofterminal displacement from its location during the lastinteraction with the network. Our model considers threedifferent velocity categories and its performance is investi-gated through a simple two-step velocity-assisted pagingalgorithm indicating certain benets in terms of pagingefciency, with possible application to multiple stepsremaining an issue for future work. It is worth mentioningthat the proposed method is very simple in terms ofcomputational complexity and could be directly combinedwith network-based methods for the estimation of terminalvelocity such as those in [1113].

2 Proposed method

For the scope of our analysis we assume that all subscribershave the same mobility prole and at any time they may bein one out of three mobile velocity categories, namely, S1, S2and S3, characterised by an average MS velocity of v1, v2and v3, respectively. These categories could, for instance,correspond to static, pedestrian and vehicular users. More-over, we assume that the users residence time in each

The authors are with the The School of Electrical, Electronic & ComputerEngineering, University of Newcastle, Newcastle-upon-Tyne, NEI 7RU, UK

r IEE, 2005

IEE Proceedings online no. 20040963

doi:10.1049/ip-com:20040963

Paper rst received 9th May 2003 and in revised form 11th May 2004

IEE Proc.-Commun., Vol. 152, No. 1, February 2005 39

velocity category follows an exponential distribution withaverage velocity category residence times m1, m2 and m3,respectively. Such an assumption has been also adopted in[14], where the authors, motivated by empirical data, haveassumed a Poisson process describing the transition betweentwo mobility categories. Finally, we assume that a userleaving S2 goes to S3 with a probability r23 and to S1 withprobability r21, with r23 +r21 1. Given the values of r21and r23, a parameter k40 can be calculated so that r23 andr21 can be expressed as r23 k/(k+1) and r211/(k+1).Arbitrary values of m1, m2, m3 and k will beassumed in this work, though more realistic values forpractical systems could be possibly derived through surveysor from the exploitation of system statistics related with theterminal velocity.The above state transition scheme constitutes a contin-

uous-time Markov chain as shown in Fig. 1.

Setting l1 1=m1; l2 1=m2 and l3 1=m3 for conve-nience, the probability rate matrix for the state transitiondiagram (Fig. 1) is given by

D l11 l12 l13l21 l22 l23l31 l32 l33

24

35

l1 l1 0l2

k 1 l2kl2k 1

0 l3 l3

264

375 1

The above matrix has three distinct eigenvalues, namely,1; 2 and 3; as follows:

1 0; 2 ap

bp

; 3 ap

bp

2where

a l1 l2 l32

3

and

b a kl1l2 l2l3k 1 l1l3 l2l3 l1l2 4

It can be shown (see Appendix) that the non-zeroeigenvalues 2 and 3 are always real and negative. As aconsequence, the probability Pij (t) can be expressed as [15]

pijt Aij Bijel2t Cijel2t 5where Pij(t) corresponds to the probability that a terminalfound in state Si at his last network access is in state Sj afterthe elapse of time t.Moreover, since the eigenvalues 2 and3 are negative, the system will eventually reach a steadystate in which the following applies:

t lim!1 Pijt Aij Pj 6where Pj corresponds to the probability of nding a user instate j without any knowledge of his previous velocitystatus. Given that Pt etL, with K, corresponding to theprobability rate matrix dened in (1), the following applies[15] for the kth order derivative of P(t):

ddt

kt0

P t: Lk 7

Taking into account (7), the coefcientsAij, Bij and Cij of (5)can be estimated for each i and j by considering the initialstates, at t 0, of P(t) and of its rst and second orderderivatives as follows:

Pij0 Aij Bij Cij 0; i 6 j1; i j

8

Pij00 l2 Bij l3Cij Li; j 9

Pij000 l22Bij l23Cij L2i; j 10Equations (9) and (10) dene a linear system of threevariables from which the calculation of Aij, Bij, and Cij isstraightforward. A typical calculation of Pij (t) for a systemwith m1 2h, m2 1h, m3 1h and k 1 is shown in Fig. 2,for i 2 and j 1, 2, 3. As expected, the relevant curvessettle to a steady state value after the elapse of a certainamount of time, which we discuss further in Section 2.2.

2.1 Estimation of expected terminaldisplacementTo estimate the displacement of the MS since its lastnetwork access, we rst need to estimate the time spent ineach velocity category. Assuming that the last networkaccess took place at t 0 and found the MS in the velocitycategory i, the expected time spent in each velocity categoryj, where i, j 1, 2, 3, until the mobile is called at time tc, isgiven by

tijtc Ztc0

Pijtdt 11

Substituting (5) into (11), we obtain

tijtc Aijtc Bijl2 el2tc 1 Cij

l3el3t 1 12

The corresponding expected displacement from originduring tc can be expressed in a worst-case sense as

Ditc ti1tcv1 ti2tcv2 ti3tcv3 13The above displacement is characterized as worst-casesince its calculation assumes no change of direction atvelocity category change.

2.2 Decorrelation time of estimatedterminal displacementIn the proposed method for the estimation of expectedterminal displacement using (13), the length of the time

11

12 23

33

32

22

21

S3S2S1

Fig. 1 Platform state transition diagram

1.0

0.8

0.6

0.4

0.2

0

prob

abilit

y

0 1 2 3 4 5 6 7 8 9 10time from last network access, h

P22

P21

P23

Fig. 2 State transition probabilities against time from last networkaccess

40 IEE Proc.-Commun., Vol. 152, No. 1, February 2005

period tc between paging and the last access of the terminalwith the network plays an important role. Indeed, for largevalues of tc the system may reach the steady state, andtherefore the benet from the exploitation of previousinformation regarding the mobile velocity category will beminimised. For the scope of our analysis, we introduce asoft threshold corresponding to the distance decorrelationtime, denoted as tmax. Considering arbitrarily the 80%decay time of each of the exponential components in (5), wehave

tmax max ln 0:2l2 ;ln 0:2

l3

14

It should be stressed that values other than the 80% of thedecay time assumed here could be used in order to adjusttmax according to the particular characteristics of a realsystem. Table 1 shows the increase of system decorrelationtime with decreased mobility. In fact, the consequence ofthis behaviour of the system is twofold. More specically,the performance of intelligent paging strategies is expectedto increase with decreased user mobility (and henceincreased system decorrelation time) and with decreasedtime between paging and last interaction with the network.This effect has been identied in previous studies, e.g. [1, 2,4], and is also intuitively expected. However, the proposedformulation and the introduction of system decorrelationtime can provide analytically a simple indication of theperformance bounds of intelligent paging schemes based onuser mobility and trafc prole, as will be discussed below.

3 Paging algorithm for exploitation of estimationof terminal displacement

The above formulation could be used as the basis for new,velocity assisted, paging strategies. As an example, wepropose a two-step velocity assisted paging (TSVAP)scheme where the location area (LA) is split into twopaging areas, namely, the rst and second paging area,denoted as FPA and SPA, respectively. We focus on thetwo-step case since in most practical systems the delayconstraints as well as the possibility of an unansweredpaging due to lack of coverage strongly discourage theimplementation of more than two paging steps. The FPAincludes all cells with central distance from the cell of lastnetwork access that is less than or equal to the distanceDi (tc) introduced in (13), multiplied by a tunable distancefactor, DF. If the paged terminal is outside the FPA, theSPA is paged, including the remaining cells. This simplealgorithm is essentially a modied form of sequential pagingincorporating the statistics of the terminal displacement asdiscussed in the preceding Section.

As far as the performance bounds of the proposed pagingmethod are concerned, we can make use of the distancedecorrelation time. For this purpose, we introduce the indexI, dened as follows:

I tmaxtc

15

Apparently, as I tends to zero, the BTS paging cost resultingfrom our proposed paging method will be maximised andwill reach that of the simultaneous paging since in this casethe MS could be located in any cell of the LA with equalprobability. On the other hand, large values of I areexpected to correspond to decreased BTS paging cost.Moreover, I could be used to predict the expected benetsfrom using TSVAP in different mobility and trafc prolecombinations, for a given LA.

4 Simulation assumption and scope

To investigate the performance of the proposed method, wehave performed a series of simulations using a simplemobility model considering two cases of LA size N, with100 and 400 cells. In all cases we have assumed square LAand cell shapes with cell size 1 1km. Subscribers areassumed to be static, pedestrian or vehicular, withcharacteristic velocities 0, 1.5 and 15m/s, respectively.During their residence in each velocity category, the MSmoves with the corresponding characteristic velocity fromthe origin. Whenever a velocity category change occurs, anew direction is selected randomly with a uniformprobability from the range [0, 2p]. With two differentassumptions regarding the average velocity categoryresidence times m1, m2 and m3 we have simulated twodifferent proles, i.e. the high and low mobility prolescorresponding to scenarios B and E of Table 1. As far as thetrafc prole is concerned, simulated MSs interact with thenetwork with a rate of l interactions per hour, Poissondistributed, covering any transaction, allowing the networkto be updated with the identity of the cell where the MS iscamped and with its velocity category (e.g. mobile-originated and mobile-terminated calls, location updates,etc.). Moreover, it is assumed that the mean call duration isnegligible compared to the time between two successivenetwork accesses. We have been focused on the valueDF 1, though one scenario has been run for DF 1.5 asan example.The performance of the TSVAP scheme is compared

with that of the simultaneous paging and also with that of atwo-step most-recent interaction area sequential paging(TSMRISP) method [2]. This latter method is considered asone of the most efcient two-step paging methods, yieldingthe minimum BTS paging cost in the absence of informa-tion related to user mobility and the unavailability of theuser location probability. It is not possible to directlycompare the performance of our proposed method withother intelligent paging schemes due to two main differencesin the fundamental assumptions. Firstly, this work does notrequire the probability of user location, and secondly itintroduces velocity category residence times which cannotbe directly converted to relevant input parameters of othervelocity-assisted paging schemes.

5 Simulation results

Our simulation results are summarised in Figs. 37. TheBTS paging cost is normalised according to its maximumvalue corresponding to the simultaneous paging method.For increasing l, the improvements obtained with the

Table 1: 80% distance decorrelation time for differentmobility scenarios, k 1

Scenario m1, m2, m3, tmax,

h h h h

A 0.5 0.5 0.5 0.8

B 1 1 1 1.6

C 2 1 1 2.32

D 1 1 2 2.32

E 3 0.5 1 2.62

F 3 1 1 2.85

IEE Proc.-Commun., Vol. 152, No. 1, February 2005 41

proposed method against TSMRISP are signicant. More-over, as expected, the BTS paging cost is lower for the lowmobility scenario compared to the high.The decreased performance of the proposed method for

low values of l is quite reasonable and can be easilyinterpreted considering its performance bounds. For thispurpose, a calculation of the index I for the two simulatedmobility cases is shown in Fig. 8. Clearly, the increase ofBTS paging cost with decreasing l for the TSVAP is in line

with the corresponding decrease of the I index of Fig. 8 forboth mobility scenarios. In fact, as can also be intuitivelydeduced, the performance of the TSVAP concerning theBTS paging cost will decrease and reach that of thesimultaneous paging as l tends to zero, since for very largevalues of time between paging and last network access theFPA will include all cells of the LA. This behaviour isevident in Figs. 5 and 6 where, for the lower values of l, theBTS paging cost of the TSMRISP is better than that of theTSVAP method.

1.2

0.8

0.4

0

no

rma

lised

BTS

pag

ing

cost

0 1 2 3 4 5 6

, network accesses/h

TSVAPTSMRISPsimultaneous

Fig. 3 BTS paging cost for low mobility caseN 100, DF 1

0

norm

alis

ed B

TS p

agin

g co

st

0 1 2 3 4 5 6

, network accesses/h

TSVAPTSMRISPsimultaneous

1.0

0.8

0.6

0.4

0.2

Fig. 4 BTS paging cost for low mobility caseN 400, DF 1

1.2

0.8

0.4

no

rma

lised

BTS

pag

ing

cost

1 2 3 4 5 6

, network accesses/h

TSVAPTSMRISPsimultaneous

00

Fig. 5 BTS paging cost for high mobility caseN 100, DF 1

1.0

0.8

0.6

0.4

0.2

no

rma

lised

BTS

pag

ing

cost

1 2 3 4 5 6

, network accesses/h

TSVAPTSMRISPsimultaneous

00

Fig. 6 BTS paging cost for high mobility caseN 400, DF 1

1.0

0.8

0.6

0.2

0.4norm

alis

ed B

TS p

agin

g co

st

1 2 3 4 5 6, network accesses/h

TSVAPTSMRISPsimultaneous

00

Fig. 7 BTS paging cost for low mobility caseN 400, DF 1.5

12

8

4

00 1 2 3 4 5

I

, network accesses/h

high, mobility scenario, low, mobility scenario,

Fig. 8 I against l for high and low mobility scenarios

42 IEE Proc.-Commun., Vol. 152, No. 1, February 2005

As far as the distance factor is concerned, Fig. 7 indicatesa further improvement obtained in the paging cost withDF 1.5 compared with the paging cost for the samescenario with DF 1, shown in Fig. 4.Finally, as far as the paging delay is concerned, results are

summarized in Fig. 9, presenting the relative paging delaydifference obtained with the TSVAP against the TSMRISPscheme. No considerable difference in the paging delay ofthe two schemes is observed, with the exception of a smallincrease of the TSVAP for the large LA case (N 400) forDF 1. However, it is interesting to observe in Fig. 9 howthis small increase in the delay is eliminated with DF 1.5,indicating that this factor could be usefully optimised. InFig. 9 we can also observe that the relative paging delaydifference between TSVAP and TSMRISP increases forincreasing l. This provides insight on the relation betweenthe BTS paging cost and the paging delay of the proposedscheme. In fact, for increasing l, the number of cellsforming the FPA of the TSVAP method decreases. Sincethe estimation of expected terminal displacement isperformed stochastically, the smaller the FPA area, themore likely is the real displacement of the terminal to begreater than the expected one, requiring a second pagingstep, thus yielding increased paging delay. However, thisminimisation of the FPA for increasing l achieved by theTSVAP method is the reason for the attainment of theconsiderable reduction in BTS paging cost that we havediscussed above.

6 Issues for future work

Since in this work we have assumed that all subscribers havethe same mobility prole, it is clearly expected that furtherimprovement in this algorithm could be obtained byidentifying velocity category residence times for eachsubscriber type (e.g. taxi drivers, house-keepers, ofceworkers etc.). This kind of information has been exploitedin [16 18] and could be investigated in conjunction withthe TSVAP. In addition, the optimisation of the distancefactor DF is also under investigation. It is expected that anoptimised setting of the DF, taking into account the usermobility pattern as well as the probability distribution of themobile velocity within each velocity category, could lead to

further improvements in the performance of the proposedmethod.

7 Conclusions

In this paper we have presented a new method for theexploitation of information related to terminal velocity andlocation during its last interaction with the network. Theproposed stochastic formulation involves three mobilevelocity categories and is exploited through a suitabletwo-step paging method. Simulation results indicate thatreduction in the BTS paging cost of up to 70% can beobtained as compared with the two-step most-recent-interaction area sequential paging method, with negligiblechange in paging delay. Issues for further investigation havealso been identied that could lead to further pagingperformance improvements.Compared to other intelligent paging schemes presented

so far, the proposed method is not based on assumptionsregarding the availability of the user location probability.Moreover, the exploitation of terminal velocity becomesmore realistic through the introduction of the velocitycategory residence times. Finally, an analytic approach isprovided regarding the performance bounds of intelligentpaging in relation to the time between the paging attemptand the last interaction of the terminal with the network,taking into account the behaviour of the user.

8 Acknowledgment

The authors wish to thank Dr John Markoulidakis(Vodafone-Panafon) and the referees of this paper for theiruseful comments and suggestions.

9 References

1 Mishra, S., and Tonguz, O.K.: Most recent interaction area andspeed-based intelligent paging in PCS. Proc. IEEE 47th VehicularTechnology Conf., 1997, Vol. 2, pp. 505509

2 Markoulidakis, J.G., and Sykas, E.D.: Performance bounds of amultiple-step paging strategy in future universal mobile tele-communications systems, Int. J. Wirel. Inf. Netw., 1995, 2, (3),pp. 133147

3 Wang, W., Akyildiz, I.F., and Stuber, G.: An optimal paging schemefor minimizing signaling costs under delay bounds, IEEE Commun.Lett., 2001, 5, (2), pp. 4345

4 Lyberopoulos, G.L., Markoulidakis, J.G., Polymeros, D.V., Tsirkas,D.F., and Sykas, E.D.: Intelligent paging strategies for thirdgeneration mobile telecommunication systems, IEEE Trans. Veh.Technol., 1995, 44, (3), pp. 543554

5 Rose, C.: A greedy method of state-based registration. Proc. IEEEInt. Conf. on Communicatons, ICC 1996, Vol. 2, pp. 11581162

6 Rose, C.: State-based paging/registration: A greedy technique, IEEETrans. Veh. Technol., 1999, 48, (1), pp. 166173

7 Tonguz, O.K., Mishra, S., and Jung, H.: A simple analyticalframework for location management in personal communicationssystems, IEEE Trans. Veh. Technol., 1998, 47, (2), pp. 428439

8 Wan, G., and Lin, E.: A dynamic paging scheme for wirelesscommunication system. Proc. ACM/IEEE Conf., Mobicom 97,Budapest, Hungary, 1997, pp. 195203

9 Mishra, S., Mahapatra, M.K.S., and Karmakar, S.: Minimization oflocation update and paging cost for PCS networks. Proc. IEEE Int.Conf. on Personal Wireless Communications, pp. 260264

10 Wong, V.W.S., and Leung, V.C.M.: Location management for next-generation personal communications networks, IEEE Netw., 2000,14, (5), pp. 1824

11 Balis, P.G., and Hinton, O.R.: Characterisation of mobile velocity inGSM using a simple feed-forward neural network, Electron. Lett.,1998, 34, (19), pp. 18101811

12 Balis, P.G., and Hinton, O.R.: Maximum likelihood differentiationbetween static and mobile users in cellular system, Electron. Lett.,2001, 37, (18), pp. 11361137

13 Hellebrandt, M., Mathar, R., and Scheibenbogen, M.: Estimatingposition and velocity of mobiles in a cellular radio network, IEEETrans. Veh. Technol., 1997, 46, (1), pp. 6571

14 Brown, T., and Mohan, S.: Mobility management for personalcommunications systems, IEEE Trans. Veh. Technol., 1997, 46, (1),pp. 269278

15 Norris, J.R.: Markov chains (Cambridge University Press, UK,1997)

30

20

10

0

-10

-20

-30

rela

tive

pagi

ng d

elay

incr

ease

, %

1 2 3 4 5

, network accesses/h

high, mobility, N =100, DF =1low, mobility, N =400, DF =1

,

high, mobility, N =400, DF =1, ,

low, mobility, N =400, DF =1.5,

low, mobility, N =100, DF =1,

Fig. 9 Relative % increase of paging delay with TSVAP comparedto TSMRISP

IEE Proc.-Commun., Vol. 152, No. 1, February 2005 43

16 Orlik, P.V., and Rappaport, S.S.: Trafc performance andmobility modeling of cellular communications with mixed plat-forms and highly variable mobilities, Proc. IEEE, 1998, 86, (7), pp.14641479

17 Markoulidakis, J.G., Lyberopoulos, G.L., Tsirkas, D.F., and Sykas,E.D.: Mobility modeling in third-generation mobile telecommunica-tions systems, IEEE Pers. Commun., 1997, 4, pp. 4151

18 Jeong, D.G., and Jeon, W.S.: Effective location management strategybased on user mobility classes. Proc. IEEE Global Telecommunica-tions Conf. GLOBECOM, 1998, Vol. 3, pp. 14261430

10 Appendix:

Taking into account (2) (4), the eigenvalues 2 and 3 willbe real and negative iff

b a k l1l2 l2l3k 1 l1l3 l2l3 l1l240 16

and

boa 17To show the validity of inequality (16) we rewrite it asfollows:

a l2kl1 l3k 1 l1l3 l2l3 l1l240 18

Assuming that l1 l3 thenk l1 l3=k 1 l3. Forthis case, therefore, by substituting the latter to (18) it

sufces to show that

a l2l3 l1l3 l2l3 l1l240 !3 l1 l2l32

4

l1l3 l1l240 ! ! l21 l22 l23 2l1l3 2l1l2 l2l340 !l1l3

l21 l22 l23 2l1l340 ! l1 l32 l2240;QED:Now for the case where l1l3, then kl1 l3=k 14l1and following the above methodology the proof of (18) andhence of (16) is straightforward.To prove (17), we need to show that

b ao0 !A1 kl1l2 l2l3k 1 l1l3 l2l3 l1l2o0 !

! l2kl1 l3k 1 ol1l3 l2l3 l1l2

19Now if l1Zl3, then kl1 l3=k 1 l1; and to prove(19) it sufces to show that l1l2ol1l3 l2l3 l1l2 !l1l3 l2l340; QED. The same methodology can befollowed to show (19) and hence (17) for the case l1ol3.

44 IEE Proc.-Commun., Vol. 152, No. 1, February 2005

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