Efficient Aroximations for Call Admission Control Performance Evaluations in Multi-Service Networks Emre A. Yavuz, and Victor C. M. Leung Deartment of Electrical and Comuter Engineering The University of British Columia Vancouver, BC, Canada V6T Z4 Email: {emrey, vleung} @ece.uc.ca Astract Several dynamic call admission control (CAC schemes for cellular networks have een roosed in the literature to reserve resources adatively to rovide the desired quality of service (QoS to not only high riority calls ut also to low riority ones. Efficient adative reservations deend on reliale and u-to-date system status feedack rovided to the CAC mechanism. However exact analysis of these schemes using multidimensional Markov chain models are intractale in real time due to the need to solve large sets of flow equations. Hence erformance metrics such as call locking roailities of various QoS classes are generally evaluated using one dimensional Markov chain models assuming that channel occuancy times for all QoS classes have equal mean values and all arriving calls have equal caacity requirements. In this aer we re-evaluate the analytical methods to comute call locking roailities of various QoS classes for several widely known CAC schemes y relaxing these assumtions, and roose a novel aroximation method for erformance evaluation with low comutational comlexity. Numerical results show that roosed method rovides results that match well with the exact solutions. I. INTRODUCTION Next generation wireless networks will suort not only voice telehony service ut also a wide variety of data services for multimedia Internet alications. Satisfying the diverse quality of service (QoS requirements of these services over cellular networks has ecome even more challenging due to reduced cell size and hence increased user moility. Call admission control (CAC schemes are deloyed to selectively limit the numer of admitted calls from each QoS class to maximize the network utilization while satisfying the QoS constraints. CAC for wired and wireless networks has een intensively studied in the ast and many riority ased CAC schemes have een roosed [] [7]. Calls with more stringent QoS requirements are given higher riorities y having exclusive access to a numer of reserved channels. Reducing locking roailities of calls with higher riorities increases the roaility of locking for calls with relatively lower riorities resulting in a trade off etween QoS classes. A set of guard channels are reserved for rioritized calls in Guard Channel (GC schemes such as cutoff riority [], fractional guard channel [2], new call ounding [3] and rigid division ased [4] schemes. Many dynamic GC schemes have also een roosed to maximize network utilization adatively This work was suorted y a grant from Bell Canada under the Bell University Laoratories rogram, and y the Canadian Natural Sciences and Engineering Research Council under grant STPGP 269872-3. [5] [7]. Efficient adative reservation deends on reliale and u-to-date erformance feedack; however exact analyses of these schemes using multidimensional Markov chain models are intractale in real time due to the need to solve large sets of flow equations [8][9]. Hence erformance metrics such as call locking roailities are generally evaluated using one dimensional Markov chain models under the simlifying assumtions that call arrivals are Poisson, channel occuancy times are exonentially distriuted with equal mean values, and traffic classes have the same caacity requirements. These assumtions may not e aroriate since calls with different riorities may have different average channel occuancy times, if not different distriutions [][]. Existing erformance evaluation methods, such as traditional and normalized, lead to significant discreancies when average channel occuancy times for distinct QoS classes are different [2]. Performance evaluation aroximation methods that have high accuracy and low comutational cost are needed if dynamic CAC schemes are to e imlemented in real time systems. In [3], Gersht and Lee roosed an iterative algorithm y modifying the aroximation suggested y Roerts [4] to imrove its accuracy when the service rates differ. However we showed in [2] that starting with an inaroriate initial value leads to significant discreancies and thus roosed a closed form aroximation method ased on one dimensional Markov chain modeling, which we called effective holding time. We assume that all classes have same caacity requirements and indeendent and exonentially distriuted channel occuancy times without the necessity of having the same average values. Numerical results showed that our aroximation method gives more accurate results when comared with the reviously roosed aroximation methods. In the asence of a roduct form solution when various classes have distinct caacity requirements, calculating the channel occuancy distriution involves solving the demanding alance equations numerically. In [5], Borst and Mitra develoed comutational algorithms for the multi-service case y couling the comutation of joint channel occuancy roailities with that of used caacity assuming that channels are occuied indeendently. The authors solved the resulting alance equations using numerical iterations. In this aer, we classify CAC schemes into two novel categories ased on the nature of communication links; symmetric and asymmetric. We define a CAC scheme as symmetric if all the communicating nodes in the state transition
(, m (, ( C m (, ( ( C m + (, C,, ( m, m, m, m, m m, ( C m no transitional flows on this side Fig.. Transition diagram for asymmetric call admission control schemes with suernodes for rioritized calls. diagram of its Markov chain model have idirectional links etween them, such as in comlete sharing (CS, comlete artitioning (CP and new call ounding schemes. We define a CAC scheme as asymmetric when the converse is true, such as in cutoff riority and fractional guard channel schemes. This aer is organized as follows. In the next section, we roose a novel erformance evaluation aroximation method for asymmetric CAC schemes. In Section III we comare the numerical results otained from the roosed method with those otained from a reviously roosed aroximation and from exact analysis. We conclude the aer in Section IV. II. PERFORMANCE EVALUATION OF ASYMMETRIC CALL ADMISSION CONTROL SCHEMES We consider a cellular system with two classes of calls: non-rioritized and rioritized calls, where the latter enjoy a higher service riority than the former. Let λ and λ denote the arrival rates, /µ and /µ denote the average channel occuancy times and and denote the required andwidth in units for non-rioritized and rioritized calls, resectively. Let C denote the total numer of channels in a cell, and q (j and q (r denote the estimated equilirium channel occuancy roailities when j rioritized calls and r non-rioritized calls, resectively, exist in the system. Let β i denote the admission roaility of an arriving non-rioritized call when the numer of usy channels is i, and k j denote the admission roaility of an arriving rioritized call when j rioritized calls exist in the cell regardless of the numer of existing non-rioritized calls. We resent the following novel erformance evaluation aroximation method, referred as state sace decomosition. Instead of evaluating the system using a one dimensional Markov chain model y grouing the nodes with the same total numer of occuied channels regardless of the tyes of call, we grou the nodes with the same numer of calls of a certain tye to otain suernodes to comose a one dimensional Markov chain model for each tye of call. By grouing nodes with the same numer of rioritized calls together to otain suernodes, as shown in Fig., we can frame a one dimensional Markov chain model that we can solve to otain the steady state roailities of each of these suernodes. The same aroach can e utilized to grou nodes that have the same numer of non-rioritized calls together as shown in Fig. 2. In Fig., we oserve that for all suernodes excet the ones that have at least one memer node that reresents a system state in which the total numer of occuied channels is equal to the total numer of channels in the system, C, there exist (m+ airs of transitional flows etween their memer nodes and the corresonding memer nodes that elong to their neighoring suernodes. Conversely, for the rest of the suernodes there exist some memer nodes that do not have transitional flows in etween any of the corresonding nodes that elong to their neighoring suernodes. Same can also e oserved for the suernodes shown in Fig. 2; however in addition to those mentioned aove there exist some other memer nodes with unidirectional transition flows. In Fig. 2, k j is the admission roaility for rioritized calls when the system is in a state that is a memer of a articular suernode j where j,,, ( C. It is similar to β i ; however, β i is a redefined user controlled arameter that indicates whether an arriving non-rioritized call will e admitted or not ased on the numer of occuied channels in the system as oosed to k j which is extracted from the multidimensional model of the system. We determine the values of the admission roailities for rioritized calls, k j, y otaining the ratio of the sum of occuancy roailities of the feasile memer nodes of a suernode, for which the system admits an arriving rioritized call, to the sum of occuancy roailities of all feasile memer nodes of that articular suernode. Thus, when j,, ( ( C ( m, the admission roailities for rioritized calls, k j, are equal to. The equilirium channel occuancy roaility when exactly j rioritized calls exist, q (j, where j,, C, can e otained from the following recursive equation. ( ρ k q ( j j q ( j, j,..., C ( j
(, m (, ( C m (, ( ( C m + (, C,, ( m, no transitional flows on this side unidirectional transition flows for non-rioritized calls ( m, m, m, m m, ( C m Fig. 2. Transition diagram for asymmetric call admission control schemes with suernodes for non-rioritized calls. Solving for q ( in the equation q ( j, we otain where q ( j j ( ρ k C j z z q(, j! j ( ( + j C ρ kn n q j j! ( C Let h r, where r,, m, denote the admission roaility of an arriving non-rioritized call when r non-rioritized calls exist, regardless of the numer of existing rioritized calls. Similar to, yet slightly different than k j, we determine the values of h r y otaining the ratio of the sum of occuancy roailities of the feasile memer nodes of a suernode, for which an arriving non-rioritized call is admitted, multilied with β i to the sum of occuancy roailities of all the feasile memer nodes of that articular suernode. The equilirium channel occuancy roailities, q (r, could e otained similarly to rioritized calls if unidirectional transition flows, shown in Fig. 2, did not exist. However their existence needs to e taken into account y adjusting µ affiliated with each suernode aroriately. Therefore we initiate µ (r to relace µ affiliated with each suernode in the model and determine its value y dividing the numer of transition flows dearting from the associated suernode with the numer of airs of idirectional transition flows in etween the same articular suernodes. ( ( C r + (2 (3 µ ( r, r,..., m (4 m ( r Then we can otain the occuancy roailities q (r, r,, ( m, which satisfy the recursive equation: ( λ h q ( r r µ ( r q ( r, r,..., m (5 where r Solving for q ( in the equation m q ( r, we otain q ( r r ( z µ ( λ hz ( r q(, r! m q ( + r r n r r m λ hn ( µ ( r r! The admission roailities for oth rioritized, k j, and non-rioritized calls, h r, cannot e otained without comuting the occuancy roaility of each feasile node. Even if the occuancy roailities of suernodes for rioritized and non-rioritized calls can e otained using this method, we still need to comute the occuancy roailities of certain feasile nodes since joint occuancy roailities of these suernodes cannot e used due to their deendencies. To overcome these difficulties, we suggest the following iterative aroach:. Initialize the value of estimated equilirium occuancy roailities ( q ˆ( n, for n,... m and n,... C y setting them equal to / (total numer of feasile nodes. 2. Calculate (r m using (4. µ for r,, 3. Iterate with the following stes until the changes in the udated values of k j, and h r are less than a chosen resolution. (6 (7
Fig. 3. Prioritized call locking roaility for the cutoff riority scheme and 3,, m 24. Fig. 4. Non-rioritized call locking roaility for the cutoff riority scheme and 3,, m 24. 3.. Calculate and udate k j for j,, ( C and q (j for j,, C using (2 and (3. 3.2. Udate the values of the estimated occuancy roailities, q ˆ( n, n, y aortioning the value of the last udated occuancy roaility, q (j, of the corresonding suernode for rioritized calls amongst its nodes with resect to the value of the last udated occuancy roaility, q (r, of the corresonding suernode for non-rioritized calls. 3.3. Calculate and udate h r for r,, ( m and q (r for r,, m using (6 and (7. 3.4. Udate the values of estimated occuancy roailities, q ˆ( n, n, y aortioning the value of last udated occuancy roaility, q (r, of the corresonding suernode for non-rioritized calls amongst its nodes with resect to the value of last udated occuancy roaility, q (j, of the corresonding suernode for rioritized calls. 4. Otain call locking roailities for rioritized and non-rioritized call using q ˆ( n, n. The call locking roailities for oth tyes of calls are calculated as follows when the final estimated values of equilirium occuancy roailities, q ˆ( n, n, are otained. m n q ( n, ( C ( n ˆ (8 m a ( C a. n qˆ ( a, n (9 ( a + n m Desite its iterative nature, we exect the state sace decomosition method to have a low comutational comlexity since decomosing the whole state sace into susaces and forming suernodes to aly one dimensional Markov chain modeling utilize the closed form formulas otained from one dimensional Markov chain models and make the roosed method easy to imlement for real time alications. III. NUMERICAL RESULTS In this section we comare the erformance of the roosed method with Borst and Mitra s aroximation [5] and the direct numerical method. We investigate the cutoff riority scheme using the following set of arameters: C 32, m 24, λ., /µ 2, /µ 5, and 3, and λ is varied from to.5. Figs. 3 and 4 deict the rioritized and non-rioritized call locking roailities, resectively, under varying rioritized call traffic load. We oserve for oth values of that when ρ > ρ, oth call locking roailities aroximated y the roosed method match the exact results (direct very well. However Borst and Mitra s method overestimates the rioritized call locking roaility generously while it underestimates the non-rioritized one extensively. When ρ < ρ, the roosed method slightly overestimates the rioritized call locking roaility while it underestimates the non-rioritized one with the discreancy increasing as oth traffic loads are decreasing. Yet, Borst and Mitra s method gives a etter aroximation only when oth traffic loads are very low due to its assumtion on indeendent channel occuancy. The discreancy oserved when ρ < ρ is due to an assumtion that we made in the iterative solution descried aove, i.e., the steady state roailities of all nodes that are memers of the same articular suernode for rioritized calls are roortional to each other with the same ratio that exists etween the steady state roailities of the corresonding suernodes for non-rioritized calls, and vice versa. When the numer of shared channels, m, is increased to 28, call locking roailities for oth tyes of calls are estimated more accurately since the corresonding transition diagram has more
Fig. 5. Prioritized call locking roaility for the cutoff riority scheme, and 3, m 28. Fig. 6. Non-rioritized call locking roaility for the cutoff riority scheme, and 3, m 28. suernodes for non-rioritized calls, each having relatively less numer of memer nodes. The following set of arameters are chosen in Figs. 5 and 6: C 32, m 28, λ., /µ 2, /µ 5, and 3, and λ is varied from to.5. The results are similar to the first case; however when increases, the discreancy oserved in the non-rioritized call locking roailities otained from the roosed method increases while it decreases for the ones otained from Borst and Mitra s method. Still, this method gives a etter aroximation than the roosed method only when oth traffic loads are very low. The comutational comlexity of the roosed method is comarale to the comlexity of a closed form solution when the required comutation time and memory are considered. The results are not resented here due to sace constraints. IV. CONCLUSION In this aer we have roosed a novel comutationally efficient aroximation method that uses an iterative aroach to evaluate the call locking erformance of asymmetric call admission control schemes in multi-service networks. Considering the high comutational comlexity of the existing numerical solution methods, we elieve that roviding erformance evaluation aroximation methods with low comutational comlexity will hel motivate the ractical imlementation of dynamic call admission control schemes in cellular moile networks. REFERENCES [] D. Hong and S. S. Raaort, Traffic model and erformance analysis for cellular moile radiotelehone systems with rioritized and norioritized handoff rocedures, IEEE Transactions on Vehicular Technology, vol. 35,. 77-92, Aug. 986. [2] R. Ramjee, R. Nagarajan, and D. Towsley, "On otimal call admission control in cellular networks," Wireless Networks, vol. 3, no.,. 29-4, March 997. [3] Y. Fang, and Y. Zhang, Call admission control schemes and erformance analysis in wireless moile networks, IEEE Transactions on Vehicular Technology, vol. 5, no.2,. 37-382, March 22. [4] M. D. Kulavaratharasah, and A. H. Aghvami, Teletraffic erformance evaluation of microcell ersonal communication networks (PCN s with rioritized handoff rocedures, IEEE Transactions on Vehicular Technology, vol. 48, no.,. 37-52, Jan. 999. [5] M. Naghshineh and S. Schwartz, Distriuted call admission control in moile/wireless networks, IEEE Journal on Selected Areas in Communications, vol. 4, no. 4,. 7-77, May 996. [6] C. Oliveira, J. B. Kim, and T. Suda, An adative andwidth reservation scheme for high-seed multimedia wireless networks, IEEE Journal on Selected Areas in Communications, vol. 6,. 858-874, Aug. 998. [7] P. Ramanathan, K. M. Sivalingam, P. Agrawal, and S. Kishore, Dynamic resource allocation schemes during handoff for moile multimedia wireless networks, IEEE Journal on Selected Areas in Communications, vol. 7, no. 7,. 27-283, July 999. [8] S. S. Raaort, The multile call handoff rolem in ersonal communications networks, IEEE 4th Vehicular Technology Conference,. 287 294, May 99. [9] S. S. Raaort and G. Monte, Blocking, hand-off and traffic erformance for cellular communication systems with mixed latforms, IEEE 42nd Vehicular Technology Conference, vol. 2,. 8 2, May 992. [] Y. Fang and I. Chlamtac, Teletraffic analysis and moility modeling for PCS networks, IEEE Transactions on Communications, vol. 47,. 62-72, July 999. [] Y. Fang, I. Chlamtac, and Y. B. Lin, Channel occuancy times and handoff rate for moile comuting and PCS networks, IEEE Transactions on Comuters, vol. 47,. 679-692, June 998. [2] E. A. Yavuz and V. C. M. Leung, Comutationally Efficient Method to Evaluate the Performance of Guard-Channel-Based Call Admission Control in Cellular Networks, IEEE Transactions on Vehicular Technology, vol. 55, no. 4,. 42-424, July 26. [3] A. Gersht and K. J. Lee, A andwidth management strategy in ATM networks, Technical reort, GTE Laoratories, 99. [4] J. W. Roerts, Teletraffic models for the Telecom integrated services network, Proceedings of the th International Teletraffic Conference, Montreal, 983. [5] S. C. Borst and D. Mitra, Virtual artitioning for roust resource sharing: comutational techniques for heterogeneous traffic, IEEE Journal on Selected Areas in Communications, vol. 6, no. 5,. 668-678, June 998.