Blocking Effects of Mobility and Reservations in Wireless Networks

Transcription

1 Blockng Effects of Moblty and Reservatons n Wreless Networks C. Vargas M. V. Hegde M. Naragh-Pour Ctr. de Elec. y Telecom Dept. of Elec. Engg. Dept. of Elec. and Comp. Engg. ITESM Washngton Unversty Lousana State Unversty Monterrey, MX St. Lous, MO 6313 Baton Rouge LA 783 Abstract Abstract We evaluate the effects of moblty and reservatons on new call blockng and handoff blockng n multrate wreless networks. The model evaluated uses Fxed Channel Assgnment (FCA) wth prorty for handoffs over new call arrvals by reservng a number of channels for handoff calls n all the cells. The performance measures used are new call blockng and handoff drop probabltes. The methodology used s that of mpled costs whch we calculate from the the network net revenue whch consders the revenue generated by acceptng a new call nto the network as well as the cost of a handoff drop n any cell. Smulaton and numercal results are presented showng the accuracy of the model. We present numercal results showng the effect of reservatons on call blockng probablty. The mpled cost analyss shows that moblty has a sgnfcant knock-on effect on the traffc elsewhere n the network and we capture ths effect through the net revenue whch s senstve to the level of moblty. We calculate the sum revenue for a gven network by maxmzng the net revenue usng mpled costs n a gradent descent algorthm. Ths analyss ndcates that n the case of multple classes of traffc the call carryng capacty of the network s senstve to the choce of reservaton parameters. I. Introducton Handoff blockng probablty s an mportant crteron n the performance of wreless networks. Typcally, on account of customer ndgnaton, the rejecton of a handoff s consdered to be more undesrable than the rejecton of new ncomng calls. Essentally three methods are used for the admsson of handoffs and new call arrvals. One treats handoff calls and new calls equally for occupancy of the channels, the second reserves channels n each cell to gve prorty to handoffs and the thrd sends handoffs to a queue f no channel s avalable. Performance evaluaton algorthms for these strateges have been ntroduced, for example n [2] for the reservaton strategy and the queueng strategy and n [6] for the reservaton and no reservaton strategy. The trade-offs between new calls and handoff calls are analyzed n [7] where a nonreservaton strategy s analyzed for very low and very hgh moblty of customers. For a treatment of handoff ssues and performance comparson of handoff strateges we refer to [5] and for the combnaton of handoff strateges and channel assgnment we refer to [8]. In general, the analyss technques used for evaluatng the performance of wreless networks requre fxed pont computatons to obtan blockng probablty and/or handoff drop probablty. The use of fxed pont computatons and the consequent mplctness of the dependence of the blockng probablty or the handoff drop probablty on the entre network traffc obscure the effects of varables such as exogenous nputs on the performance measures. In ths paper we use the concept of mpled cost [3], [4] to evaluate these. We assume Fxed Channel Allocaton (FCA), (.e., every cell s assgned a fxed number of channels) wth reservaton to gve prorty to handoff calls. II. Model for Multrate Wreless Networks Consder an asymmetrc cellular network wth fxed channel assgnment where N s the set of cells and N, the total number of cells. Each cell has C channels assgned to t. There are M classes of traffc whch share the network resources. Let b m be the number of channels requred by traffc of class m, m = 1, 2,...,M, < b 1 b 2 b M. The new call arrval process of class m to cell s a Posson process wth mean λ,m ndependent of other new call arrval processes. The tme a call of class m remans n cell, the dwell tme, s a random varable wth exponental dstrbuton and mean 1/µ,m and t s ndependent of earler arrval tmes, call duratons and elapsed tmes of other users. At the end of a dwell tme a call may attempt a handoff to an adjacent cell, reman wthn the cell or leave the network. Let q j,m be the probablty that a call of type m n progress n cell after completng ts dwell tme goes to cell j,.e., there s a class m handoff from cell to cell j, let q T,m be the probablty that a call of class m n progress n cell after completng ts dwell tme termnates and abandons the network and let q,m be the probablty that a call of class m n progress n cell after completng ts dwell tme remans wthn cell. If cell and cell j are not adjacent then q j,m =for m =1,...,M. All the cells have a channel reservaton parameter, T m, for calls of class m. The reservaton pa-

2 rameters are ntended to gve prorty to handoff calls wth respect to new calls through a reservaton polcy as descrbed below. Let n = (n 1,n 2,...,n M ) be the current state of cell where n m s the number of calls of class m present n cell. Defne U (m) and B (m) as the set of unreserved and blocked states for traffc class m n cell, respectvely, where U (m) k=1 M C n k b k = {n : T m < b m } and { } B (m) = n : C M n k b k <b m, and where x s k=1 the largest nteger less than or equal to x. Clearly B (m) Q (m). The reservaton polcy can be stated as follows. If a new call of class m arrves to cell, ts accepted f the state of cell, n, snu (m), otherwse, t s blocked. If a handoff call of class m arrves to cell, t s blocked only f n B (m). We consder that occupancy of the cells evolves accordng to an M-dmensonal brth-death process ndependent of other cells. We note however that the transton probabltes of ths process depend on the steady state probabltes of the other cells. Let p (n) be the steady-state probablty that cell s n state n. From p (n), the new call blockng probablty for calls of class m n cell, B,m, and the handoff drop probablty for calls of class m n cell, B h,m,canbe calculated as follows: B,m = p (r), and B h,m = p (r). r Q (m) r B (m) In [1] we show that the forced termnaton probablty, Bj,m d, vz., the probablty that a call of class m whch orgnated n cell j s termnated due to a handoff falure durng ts lfetme can be calculated from these steady-state probabltes as well. III. Impled Costs Defne the net revenue, W, as the revenue generated by the traffc whch s carred successfully. Ths revenue conssts of two components: the frst one s the revenue, λ j,m (1 B j,m ) w j,m, generated by acceptng n each cell j a new call of class m and the second component, λ j,m (1 B j,m ) c j,m Bj,m d, takes nto account the cost of a forced termnaton due to handoff falure of those new calls of class m that have been accepted n cell j. Here w j,m s the revenue generated by acceptng a call of class m n cell j, andc j,m s the cost of a forced termnaton of a call of class m due to a handoff falure. Hence the net revenue s W (λ) = λ j,m (1 B j,m (λ, p)). m=1 j N { wj,m c j,m Bj,m(λ, d p) }, where p denotes the vector whose components are the steady-state probabltes for each state of all the cells. and λ, the vector of new call arrval rates. The revenue for cell and traffc class m, w,m,s proportonal to b m and the average holdng tme of the calls of that class. The average holdng tme depends on the average number of handoffs the calls undergo before departure from the network and snce every tme a call s accepted n a cell ts duraton n that cell s the dwell tme wth mean 1/µ,m, the average holdng tme of a call wll be gven by the average sum of the dwell tmes that the call wll undergo before ts departure from the network. In order to reflect the greater loss of servce caused by handoff blockng, c,m s chosen to be hgher than W,m by a unt. The fxed pont model descrbes p as an mplct functon of λ. TheB j,m,b hj,m and Bj,m d are, n turn, functons of p and thereby mplct functons of λ. Consequently, W (λ) s also an mplct functon of λ. We undertake a careful and extensve effort to obtan relatons of total and partal dervatves of the new call and handoff blockng probabltes by dfferentatng the fxed pont equatons. In partcular we calculate the total dervatve of the net revenue functon wth respect to new call arrval rates, dw (B,B h,λ,α) dλ k,m. As a result of the analyss we are able to obtan rates of change between dfferent performance measures, between resources n one part of the network and performance n another part, knock-on effects of a change n traffc or resource, all of whch are very useful n the desgn and performance evaluaton of these networks. IV. Numercal Results Ths secton presents numercal and smulaton results for the model presented n prevous sectons. We verfy the accuracy of the fxed pont model, evaluate network revenue for varous cases of load and moblty, present examples of how the mpled costs can be ncorporated nto call prcng and obtan the sum revenue for a network wth dfferent parameters. A. Blockng and Revenue Results The calculatons and smulatons were done for the 1-cell network shown n Fgure 1 wth two classes of traffc. In the fgures, T =[x, y] refers to the reservaton of x channels for calls of class 1 and y channels for calls of class 2 n all the cells, and µ =[x, y] refers to a dwell tme of 1/x for calls of class 1 and 1/y for calls of class 2 n all the cells. The number of channels needed by each class of traffc are b 1 =1andb 2 =2. The departure probabltes are q T,1 =.8andq T,2 =.85. These parameters are summarzed n Table

3 I. The parameters q j,m are chosen as (1 qt,1) L where L s the total number of cells the call could move to ncludng the current cell. In the examples, the channel reservaton parameters of the classes were vared and the performance evaluated n terms of the new call blockng probablty, B,m, and the handoff drop probablty, B h,m,ofboth classes. The fgures shown were obtaned by keepng the new call arrval for all the cells and both classes constant except for calls of class 1 n cell 1 snce ths s one of the cells wth the most number of adjacent cells. The total new call arrval rate nto a cell for both classes s 2% of the capacty of that cell. Fgures 2-4 contan graphs of comparson of the numercal results for new call blockng and handoff blockng wth the smulatons for dfferent channel reservaton parameters. In all cases t can be seen that smulaton results are extremely close to the numercal results, thereby valdatng the fxed pont model. It can be seen that for both classes of calls, the new call blockng for the case of no reservaton performs better than the case wth reservaton (for handoff drop probablty, when channels are reserved, the performance mproves). Ths mprovement s due to the prorty gven to handoffs by reservng channels n each cell. In the case where there s no reservaton, Fgure 2, the new call blockng and the handoff drop probabltes are the same. Fgure 5 shows the net revenue for the ten cell network for 5% load,.e., the new call arrval rate s 5% of the capacty of that cell. The moblty was vared between low and hgh: low moblty s characterzed by q T,1 =.8 andq T,2 =.85and hgh moblty s characterzed by q T,1 =.5 andq T,2 =.55. It can be seen that due to the ncrease n traffc n the network, because users tend to have longer connecton, tmes the blockng of new and handoff calls wll ncrease and the net revenue wll decrease. For the hgh moblty case, we have less revenue, than that wth low moblty. However n the former case reservaton can mprove the revenue. Ths can be seen n the fgure when we compare the net revenue of the no reservaton case, T =[, ], wth that of T =[1, 1], or T =[, 2]. We can also see n the fgure that for hgh moblty, the case of T =[2, 2] wll have better revenue than the no reservaton case for class 1 new call arrvals above 6 calls per tme unt. For the low moblty case, the case of no reservaton results n the hghest revenue. In Fgure 6, we show the mpled cost as a functon of new call arrval rate of calls of class one nto cell 1. It can be observed that mpled cost s a measure of the rate of ncrease of the net revenue. The larger the mpled cost the larger the rate of ncrease of revenue and the better the blockng level the network s experencng. The fgure also shows that reservaton has to be chosen wsely dependng on the mpled cost obtaned: f we have a small value of the mpled cost, a connecton establshment wll be more expensve and t s not a good dea to accept more traffc of that class n that cell. It can be seen that, for hgh moblty and class 1 new call arrval above fve calls per tme unt, the mpled cost of all the cases of reservaton analyzed decrease below the T =[2, ] case, meanng that revenue for ths reservaton wll be better than that of the other cases. In the low moblty case we can see that t wll be better to use the reservaton T =[1, 1] snce t wll represent an mprovement n revenue. B. Sum Revenue Impled costs capture the effect of ncreases n new call arrvals n one cell on the entre network. As a result, they are useful n optmzng network-wde goals. Defne the sum revenue as the maxmum sum of new call arrval rates such that the new call blockng probablty of each cell s less than or equal to some prespecfed maxmum blockng probablty. The noton of sum revenue s smlar to that of sum capacty for crcut swtched networks. Sum capacty was ntroduced and calculated for adaptve routng schemes n [9], and [1]. where mpled costs were used to solve a nonlnear constraned optmzaton algorthm. As an llustraton of the use of mpled costs n optmzng network-wde goals, we use them to calculate the sum revenue. To ths end, we formulate a constraned nonlnear optmzaton problem wth the objectve functon beng the network net revenue and constrants beng the new call blockng and handoff blockng probabltes. The ndependent varables are the new call arrval rates. Let η and γ be vectors whose components represent the maxmum new call and handoff blockng probabltes, respectvely, for each cell and let be the zero vector. Then the optmzaton problem s: max W (B, B h,λ,α)= (1) λ w,m λ,m (1 B,m (λ, p)) m=1 N c,m B h,m (λ, p) { I {Tm>}α,m (v) m=1 N + I {Tm=} [ρ,m (λ, v) λ,m ] }, subject to B η, B h γ, λ. (2) where I(.) s the ndcator functon. The soluton for the above optmzaton problem gves the maxmum revenue that the network can generate for a

4 gven blockng probablty vector. The optmzaton s acheved by usng the mpled costs n a gradent descent algorthm that gves the drecton n whch the vector of new call arrval rates has to be vared to get the desred maxmzaton. In Fgure 7 the sum revenue of the 1-cell network wth two classes of customers s shown for several values of channel reservaton parameters and for low moblty. The horzontal axs s the new call blockng of class 2 snce ths s the class wth hgher bandwdth requrement and ts new call blockng s hgher than the new call blockng probablty of class 1 and the handoff drop probablty of both classes. The best performance was obtaned for the case of T =[1, ], where there s one channel reserved for handoffs of class 1. The poorest performance was from the case T =[, 1] and the case T =[1, 1], has no sgnfcant dfference wth the T =[, 1] case. The second best case was T =[, ] wth no channels reserved for any class. It can be seen that reservaton parameters need to chosen carefully n the multrate case. It can also be concluded that ncreasng the channel reservaton for the class wth less bandwdth requrement mproves the sum revenue, whereas ncreasng t for the other class wll degrade the performance. V. Conclusons We descrbed the calculaton of mpled costs wth respect to the new call arrval rates for wreless networks wth multple classes of customers, and show ther use for evaluatng trade-offs between new call blockng and handoff blockng, and between low moblty and hgh moblty traffc. Comparson of the sum revenue ndcates that the optmzaton usng mpled costs results n a sgnfcant mprovement. Ths provdes evdence that matchng capacty dstrbuton to exogenous traffc and moblty can result n sgnfcant benefts to the network. The sum revenue n the case of multple classes of traffc ndcates the need for careful choce of reservaton parameters. Acknowledgements The second author would lke to acknowledge the support of NSF through grant NCR References [1] Vargas C., Shadow Prces for Wreless and Wrelne Networks, PhD Thess, Department of Electrcal and Computer Engneerng, Lousana State Unversty, Baton Rouge, Lousana, July [2] Hong, D., and Rappaport S., Traffc Model and Performance Analyss for Cellular Moble Rado Telephone Systems wth Prortzed and Nonprortzed Handoff Procedures, IEEE Trans. on Vehcular Tech., vol. VT-35, No. 3, August [3] Kelly, F.P., Routng n Crcut-Swtched Networks: Optmzaton, Shadow Prces and Decentralzaton, Advances n Appled Probablty, vol. 2, pp , [4] Kelly, F.P., Routng and Capacty Allocaton n Networks wth Trunk Reservaton. Mathematcs of Operatons Research, vol. 15, pp , Nov 199. [5] Ln, Y-B., Mohan, S. and Noerpel, A., Analyzng the Trade-off between Implementaton Costs and Performance: PCS Channel Assgnment Strateges for Hand-off and Intal Access, IEEE Personal Communcatons Magazne, Thrd Quarter, [6] McMllan, Davd, Traffc Modellng and Analyss for Cellular Moble Networks, Proceedngs of the ITC-13, pp , [7] Sd, M., and Starobnsk, D., New Call Blockng versus Handoff Blockng n Cellular Networks, IEEE INFOCOM 96, pp , March [8] Teknay, S. and Jabbar, B., Handover and Channel Assgnment n Moble Cellular Networks, IEEE Communcatons Magazne, November [9] Vargas, C., Hegde, M., Naragh-Pour, M. and Mn, P., Impled Costs for LLR and ALBA, IEEE/ACM Transactons on Networkng, vol. 4, no. 5, pp , October [1] Vargas, C., Hegde, M., Naragh-Pour, M. and Mn, P., Shadow Prces for State Dependent Routng, The Twenty-Eghth Annual Conference on Informaton Scences and Systems, Prnceton Unversty, Prnceton, N.J. 8544, March 1994, pp TABLE I Parameters Ten-Cell Network, 2% Load, (λ 1,1 vared) Smulaton Parameters Cell λ,1 λ,2 C µ,1 µ,2 q T,1 q T,2 1 * Cell Number Capacty Fg. 1. Ten-Cell Network Used n Examples wth Parameters for Sngle Rate Case

5 9 New Call Blockng Cell Numercal Smulaton Net Revenue (revenue per tme unt) Low Moblty Hgh Moblty... T=[,] T=[,2] T=[2,] T=[2,2].5 65 Fg. 2. New call blockng, T =[, ], µ =[1, 1 ], 2% load 2 Fg. 5. Net revenue for 1-cell network, 5% load, µ =[1, 1 2 ] Numercal Smulaton 1.8 Low Moblty... T=[,] T=[,2] T=[2,] T=[2,2] New Call Blockng Cell Impled Cost.6.4 Hgh Moblty.5 Fg. 3. New call blockng, T =[1, 1], µ =[1, 1 ], 2% load 2 Fg. 6. Impled cost for 1-cell network, 5% load, T =[, ], µ =[1, 1 2 ].3 5 Numercal Smulaton 8 75 Handoff Blockng Cell Fg. 4. Handoff blockng probablty, T =[1, 1], µ =[1, 1 ], 2% 2 load Sum Revenue (calls per tme unt) T=[,] T=[,1] T=[1,] New Call Blockng Fg. 7. Sum revenue for the 1-cell network, low moblty