Efficient application of these ... call blocking probabilities of various traffic classes for several .... demand of a packet-switched traffic stream so that application.
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Efficient Methods for Performance Evaluations of Call Admission Control Schemes in Multi-Service Cellular Networks Emre Altug Yavuz, Member, IEEE, and Victor C. M. Leung, Fellow, IEEE
Abstract—Many dynamic call admission control (CAC) schemes have been proposed in the literature for adaptive reservations in cellular networks. Efficient application of these schemes requires reliable and up-to-date feedback of system performance to the CAC mechanism. However, exact analyses of these schemes in real time using multi-dimensional Markov chain models are challenging due to the need to solve large sets of flow equations. One dimensional Markov chain models have been widely used to derive performance metrics such as call blocking probabilities of multiple traffic classes assuming that all classes of calls have equal capacity requirements and exponentially distributed channel holding times with equal mean values. These assumptions need to be relaxed for a more general evaluation of CAC performance in multi-service cellular networks. In this paper we classify CAC schemes according to their Markov chain models into two categories: symmetric and asymmetric, and develop computationally efficient analytical methods to compute call blocking probabilities of various traffic classes for several widely known CAC schemes under relaxed assumptions. We obtain a product form solution to evaluate symmetric schemes and propose a novel performance evaluation approximation method with low computational cost for asymmetric schemes. Numerical results demonstrate the accuracy and efficiency of the proposed method. Index Terms—Call admission control (CAC), call blocking probability, computational cost, performance evaluation, resource management, multi-service, cellular networks.
TABLE I N OMENCLATURE C c K m λnp λp 1/μnp 1/μp 1/μef f bnp bp ρnp ρp nnp np Bnp Bp βi kj hr qp (j) qnp (r) qˆ(c) qˆ(nnp , np )
I. I NTRODUCTION
N
EXT generation wireless networks are promising to provide not only conventional voice services but also the efficiency and flexibility of multiplexing a wide variety of traffic from data to multimedia applications. However, satisfying the diverse quality of service (QoS) requirements of these services over cellular networks are becoming even more challenging due to reduced cell size and increased user mobility. Call admission control (CAC) schemes are deployed to selectively limit the number of admitted calls from each traffic class to maximize network utilization while satisfying the QoS constraints [1]. CAC for wired and wireless networks has been
Manuscript received March 9, 2007; revised September 25, 2007; accepted October 15, 2007. The associate editor coordinating the review of this paper and approving it for publication was H. Chen. This paper is based in part on a paper presented at IEEE GLOBECOM, San Francisco, CA, November, 2006. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant STPGP 269872-03. The authors are with the Department of Electrical and Computer Engineering, The University of British Columbia Vancouver, BC, Canada V6T 1Z4 (e-mail: {emrey, vleung}@ece.ubc.ca). Digital Object Identifier 10.1109/TWC.2008.070280.
Number of channels in a cell. Each channel = 1 bandwidth unit. Number of occupied channels in a cell. Threshold in no. of channels for new call bounding scheme. Threshold in no. of channels for cutoff priority scheme. Arrival rate for non-prioritized calls. Arrival rate for prioritized calls. Average channel holding time for non-prioritized calls. Average channel holding time for prioritized calls. Average effective channel holding time. Capacity requirement in bandwidth units for non-prioritized calls. Capacity requirement in bandwidth units for prioritized calls. Traffic intensity for non-prioritized calls (i.e., λnp /μnp ). Traffic intensity for prioritized calls (i.e., λp /μp ). Number of non-prioritized calls in the system. Number of prioritized calls in the system. Blocking probability for non-prioritized calls. Blocking probability for prioritized calls. User defined admission probability for non-prioritized calls in a call admission control scheme. Admission probability for prioritized calls. Admission probability for non-prioritized calls. Equilibrium channel occupancy probability when j prioritized calls exist in the system. Equilibrium channel occupancy probability when r non-prioritized calls exist in the system. Estimated equilibrium channel occupancy probability when c channels are occupied. Estimated equilibrium channel occupancy probability for nnp non-prioritized and np prioritized calls.
extensively studied in the past and many priority based CAC schemes have been proposed [2] - [14]. One way to give calls with a more stringent QoS requirement a higher priority is to give them exclusive access to a number of reserved channels, thus reducing their call blocking probability. However, this increases the blocking probability for calls with a relatively lower priority resulting in a tradeoff between traffic classes. The goal is to sustain a balance between traffic classes while satisfying the respective QoS requirements. A set of guard channels are reserved for prioritized calls in Guard Channel (GC) schemes such as cutoff priority [2] - [5], fractional guard channel [6], new call bounding [7] and rigid division based [8] schemes. Many dynamic GC schemes have also been proposed to maximize network utilization adaptively [9] - [14]. Efficient adaptive reservation depends on reliable and up-to-date performance feedback; however, exact analyses of these schemes using multi-dimensional (MD) Markov chain
c 2008 IEEE 1536-1276/08$25.00
YAVUZ and LEUNG: EFFICIENT METHODS FOR PERFORMANCE EVALUATIONS OF CALL ADMISSION CONTROL SCHEMES
(MC) models are intractable in real-time due to the need to solve large sets of flow equations. Hence, performance metrics such as call blocking probabilities are generally evaluated using one-dimensional (1D) MC models under the simplifying assumptions that call arrivals are Poisson, and that all calls have the same capacity requirements and exponentially distributed channel holding times with equal mean values regardless of the traffic classes they belong to. Due to the popularity of Internet and multimedia applications, increasingly traffic carried over wireless networks is packet-switched and statistically multiplexed over shared channels to improve network utilization, which makes performance evaluation of CAC schemes harder due to the dynamic nature of the traffic. This difficulty can be overcome by using the effective bandwidth [15] - [17] to represent the traffic demand of a packet-switched traffic stream so that application of the above CAC schemes to a packet-switched network can still be evaluated using MC models. This approach has been successfully applied to cellular networks [16]. However, the simplifying assumptions mentioned above may not be appropriate in many situations since channel holding times of calls with different priorities may have different average values if not different distributions [18] - [19]. Existing performance evaluation methods based on 1D MC approximations lead to significant discrepancies when average channel holding times for distinct traffic classes are different [20]. Thus exact analysis methods based on MD MC models appear to be the only means to obtain accurate solutions for evaluating CAC schemes. In [21], Rappaport obtained call blocking probabilities for calls of various priorities in a cellular network by using a MD model, and with Monte in [22], developed an analytical model for traffic performance analysis using a MD birth-death process to take into consideration the effects of various platform types distinguished by different mobility characteristics. Schembra introduced two MD models in [23] to assess the effectiveness of a proposed channel management strategy with respect to both network utilization and user performance. These methods suffer from the curse of dimensionality, which results in very high computational cost for large systems, despite providing the exact solutions. Approximation methods that have a high accuracy and low computational cost are needed if dynamic CAC schemes are to be implemented in real-time systems that adapt to dynamic changes in traffic statistics. Li and Chao obtained a product form solution to study cellular networks with a single class of calls in [24] and multiple classes of calls in [25] by modeling a multicell network as a network of queues employing a hybrid GC/QP (queuing priority) scheme with transfer of unsuccessful requests to neighboring cells; however, their solution is restrictive to the protocol considered and may be difficult to extend to multi-service models. Gersht and Lee proposed an iterative algorithm [26] by modifying Roberts’ approximation [27] to improve its accuracy when the service rates differ, but this algorithm can suffer significant discrepancies from inappropriate initial values [20]. Yavuz and Leung proposed an easy-to-implement closed form approximation method [20] based on 1D MC modeling, which assumes that all calls have the same capacity requirements and independent exponentially distributed channel holding times that do not necessarily have
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the same average values. Yet this method is useful only when the call traffic is homogeneous in terms of capacity requirements. When call traffic is heterogeneous in capacity requirements for different traffic classes, owing to the absence of a product form solution, calculating the channel occupancy distribution is demanding for all but the smallest systems as it involves solving the balance equations numerically. Borst and Mitra [28] developed computational algorithms for the multiservice case by coupling the computation of joint channel occupancy probabilities with that of used capacity assuming that channels are occupied independently, and solving the resulting balance equations using numerical iterations. However, due to the independence assumption on channel occupancy, these algorithms give accurate results only when a large number of call-types with different capacity requirements are involved. In this paper, we consider multiple traffic classes with different capacity requirements, and classify prioritized CAC schemes into two categories: symmetric and asymmetric, based on the nature of connecting links in a scheme’s two-dimensional MC state transition diagram. We present performance evaluation methods with low computational costs for each category under the simplifying assumptions that call inter-arrival times and channel holding times for all traffic classes are exponentially distributed, but with different average values in general. This paper is organized as follows. In the next section we obtain the product form exact solution formula to evaluate symmetric CAC schemes in multi-service networks. In Section III, we propose a novel approximation method to evaluate asymmetric CAC schemes in multi-service networks. Section IV presents numerical results to compare the approximation method proposed in section III with the exact analysis and previously proposed approximation methods. We show that the computational cost of the proposed approximation method is significantly lower than that of the exact analysis. Section V concludes the paper. II. P ERFORMANCE E VALUATION OF S YMMETRIC C ALL A DMISSION C ONTROL S CHEMES We define a CAC scheme as symmetric if each pair of nodes in the state transition diagram of the scheme’s MC model commute bi-directionally if they are connected. The widely known complete sharing (CS), complete partitioning (CP) and new call bounding schemes can be regarded as symmetric. In this section we obtain an exact product-form solution to evaluate symmetric CAC schemes in multi-service networks where all traffic classes have distinct bandwidth requirements. We consider a cellular system employing the new call bounding scheme to serve two classes of calls with different bandwidth requirements: non-prioritized and prioritized, where the latter enjoy a higher service priority than the former. This scenario is chosen for the benefit of simplicity although the analytical method could accommodate any number of classes and the other CAC schemes mentioned above. We assume that the arrival processes for non-prioritized and prioritized calls are Poisson, with respective arrival rates λnp and λp , and that their channel holding times are exponentially distributed with average channel holding times 1/μnp and 1/μp , respectively. Let C denote the total number of channels in a cell and bnp and bp denote the required bandwidth (in bandwidth units, where 1
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channel = 1 bandwidth unit) for non-prioritized and prioritized calls, respectively. In practice C is technology dependent and may vary from cell to cell, e.g., in interference-limited CDMA systems. Nevertheless, we assume for simplicity that the value of C is fixed and known for each cell. Let integer K(0 ≤ K ≤ C) be the threshold for the new call bounding scheme; then a non-prioritized call is admitted only when less than K channels are occupied by non-prioritized calls and the total number of occupied channels is less than the total number of channels, C, in the cell. The traditional method for CAC performance evaluation, which uses a 1D MC model with a single fixed average channel holding time for all cell traffic, leads to inaccurate results when different traffic classes with diverse average channel holding times exist. A product form solution is presented in [7] to accurately obtain the blocking probability Bnp of non-prioritized calls, and blocking probability Bp of prioritized calls, by exploiting the symmetric nature of the scheme assuming that all traffic classes have the same capacity requirements. In [7], the average channel holding times for both types of calls are normalized to allow the arriving traffic for each type of call to be scaled appropriately. Here, we extend that product form solution in [7] so that all traffic classes can have distinct capacity requirements. Let traffic loads ρnp = λnp /μnp and ρp = λp /μp , then an equivalent MC model, in which the prioritized and nonprioritized call arrivals are Poisson with arrival rates ρp and ρnp , respectively, and service rates (reciprocal of average channel holding times) of all calls are equal to 1, can be used to give a simplified representation of the system since only traffic loads are required to obtain the stationary distributions. Let q(nnp , np ) denote the steady state probability that there are nnp non-prioritized calls and np prioritized calls in the system. Then we obtain the following stationary distribution: ρnp nnp ρp np · · q(0, 0) (1) q(nnp , np ) = nnp ! np ! 0 ≤ nnp .bnp ≤ K, (nnp .bnp + np .bp ) ≤ C, nnp ≥ 0, np ≥ 0 where ⎤−1 ⎡ nnp np ρ ρ np p ⎦ · q(0, 0) = ⎣ nnp ! np ! (nnp ,np )∈S ⎤−1 ⎡ K/bnp ·bnp ))/bp np ρnp nnp (C−(nnp ρ p ⎦ · =⎣ (2) n ! n ! np p n =0 n =0 np
p
and xis the ”floor” function that rounds argument x to the largest integer less than or equal to x. Thus, the formulas for non-prioritized and prioritized call blocking probability are as follows: K/bnp (C−(nnp ·bnp ))/bp
Bnp =
nnp =0
np =0
q(nnp , np )
K/bnp (C−(nnp ·bnp ))/bp
=
nnp =0
np =0 ρnp nnp nnp !
·
ρp np np !
K/bnp ρnp nnp (C−(nnp ·bnp ))/bp ρp np nnp =0 np =0 nnp ! · np !
(3)
K/bnp
Bp =
q(np , (C − (np · bnp ))/bp ) =
np =0 n
K/bnp
np =0
ρ ρnp np !
·
((C−(np ·bnp ))/bp ) ρp ((C−(np ·bnp ))/bp )!
K/bnp ρnp nnp nnp =0 nnp !
·
(C−(nnp ·bnp ))/bp np =0
ρp np np !
(4)
{(np .bp + nnp .bnp ) ≥ C} ∪ {nnp .bnp ≥ K} When K = C, the new call bounding scheme becomes the non-prioritized scheme; however, the blocking probabilities for non-prioritized calls, Bnp , and prioritized calls, Bp , will not be the same unless bnp = bp . When bnp = bp = 1, both probabilities become: (ρnp +ρp )C C! (ρnp +ρp )j j=0 j!
Bnp = Bp = C
(5)
III. P ERFORMANCE E VALUATION OF A SYMMETRIC C ALL A DMISSION C ONTROL S CHEMES We define a CAC scheme as asymmetric when some pairs of nodes in the state transition diagram of the scheme’s MC model have only unidirectional links connecting them. Both the widely known cutoff priority and fractional guard channel schemes, which decide on whether an arriving non-prioritized call is accepted or not based on the number of total occupied channels in the system, can be regarded as asymmetric schemes. We consider a cellular system with two classes of calls where prioritized calls enjoy a higher service priority than non-prioritized ones. Let λp ,λnp , μp , μnp , bp , bnp , and C be defined as before and qp (j) and qnp (r), respectively, denote the estimated equilibrium channel occupancy probabilities when exactly j prioritized calls and r non-prioritized calls exist in the system. Let βi (i = 0, 1, . . . , C − 1) denote the admission probability of an arriving non-prioritized call when the total number of busy channels is i. In the fractional guard channel scheme, 0 ≤ βi ≤ 1 and βi decreases as i increases, whereas in the cutoff priority scheme, βi = 1 for i = 0, 1, . . . , m − 1 and βi = 0 otherwise, where m is the cutoff threshold. Let kj (j = 0, 1, . . . , C/bp − 1) denote the admission probability of an arriving prioritized call when j prioritized calls exist in the cell regardless of the number of existing non-prioritized calls. Thus kj is similar to βi ; however, βi is a predefined user-controlled parameter that indicates whether an arriving non-prioritized call will be admitted or not based on the number of occupied channels in the system, whereas kj is extracted from the MD MC model of the system. We present the following novel performance evaluation approximation method referred herein as state-space decomposition. Instead of evaluating the system using a 1D MC model by grouping nodes with the same total number of occupied channels regardless of the types of calls, we group nodes with the same number of calls of a certain type to form ”supernodes”, thus resulting in a different 1D MC model for each type of call. Consider the cutoff priority scheme as an example. By grouping nodes with the same number of prioritized calls such as (0, 0), (bnp , 0), . . . , (bnp · (m/bnp − 1), 0), (bnp · (m/bnp ), 0) or (0, bp ), (bnp , bp ), . . . , (bnp ·
YAVUZ and LEUNG: EFFICIENT METHODS FOR PERFORMANCE EVALUATIONS OF CALL ADMISSION CONTROL SCHEMES (0, b p .⎣m / b p ⎦)
(0, b p .⎣(C − m) / b p ⎦)
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(bnp .⎣m / bnp ⎦,0)
no transitional flows on this side (bnp .⎣m / bnp ⎦, b p )
Fig. 1.
(bnp .⎣m / bnp ⎦, b p .⎣m / b p ⎦)
(bnp .⎣m / bnp ⎦, b p .⎣(C − m) / b p ⎦)
Transition diagram for asymmetric call admission control schemes with supernodes for prioritized calls. (0, b p .⎣m / b p ⎦)
0,0
(0, b p .⎣(C − m) / b p ⎦)
(0, b p .( ⎣(C − m) / b p ⎦+ 1))
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0, b p
no transitional flows on this side (bnp .( ⎣m / bnp ⎦− 1),0)
unidirectional transitional flows for non-prioritized calls
(bnp .⎣m / bnp ⎦,0)
Fig. 2.
(bnp .⎣m / bnp ⎦, b p )
(bnp .⎣m / bnp ⎦, b p .⎣m / b p ⎦)
(bnp .⎣m / bnp ⎦, b p .⎣(C − m) / b p ⎦)
Transition diagram for asymmetric call admission control schemes with supernodes for non-prioritized calls.
(m/bnp − 1), bp ), (bnp · (m/bnp ), bp ) together to obtain the supernodes shown in Fig. 1, we can frame a 1D MC model that we can solve to obtain the steady state probability of each of these supernodes. The same approach can be utilized to group nodes that have the same number of nonprioritized calls such as (0, 0), (0, bp ), . . . , (0, bp · (C/bp )) or (bnp , 0), (bnp , bp ), . . . , (bnp , bp ·((C − bnp )/bp )) together. The different 1D MC models obtained above are given in Figs. 1 and 2 for prioritized and non-prioritized calls, respectively. In Fig. 1, we observe that for all supernodes except the ones that have at least one member node that represents a system state in which the total number of occupied channels is equal to the total number of channels in the system, C, there exist (m + 1) pairs of transitional flows between their member nodes and the corresponding member nodes that belong to their neighboring supernodes. Conversely, for the rest of the
supernodes there exist some member nodes that do not have transitional flows in betwen any of the corresponding nodes that belong to their neighboring supernodes. The same can also be observed for the supernodes shown in Fig. 2; however, in addition to those mentioned above there exist some other member nodes with unidirectional transition flows. In Fig. 3, we show the 1D MC model for prioritized calls where each node represents a supernode composed of a set of nodes shown in Fig. 1. We determine the values of the admission probabilities for prioritized calls, kj , by obtaining the ratio of the sum of occupancy probabilities of the feasible member nodes of a supernode, for which the system admits an arriving prioritized call, to the sum of occupancy probabilities of all feasible member nodes of that particular supernode. Thus, when j = 0, 1, . . . , ((C − (m/bnp · bnp )/bp − 1), admission probabilities for prioritized calls, kj , are equal to 1.
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The equilibrium channel occupancy probability when exactly j prioritized calls exist, where qp (j), j = 0, 1, . . . C/bp , can be obtained recursively from the following equation. (ρp · kj−1 ) · qp (j − 1) = j · qp (j), j = 1, . . . , C/bp (6) C/bp Solving for qp (0) in the equation j=0 qp (j) = 1, we obtain
j−1 (ρp · kz ) · qp (0), 1 ≤ j ≤ C/bp qp (j) = z=0 (7) j! where
⎡ qp (0) = ⎣1 +
j−1 (ρp · kn )
C/bp
n=0
j=1
j!
⎤−1 ⎦
(8)
Let hr , where r = 0, 1, . . . , (m/bnp − 1), denote the admission probability of an arriving non-prioritized call when r non-prioritized calls exist, regardless of the number of existing prioritized calls. Note that hr should not be confused with βi since the latter is a predefined user-controlled parameter based on the total number of busy channels. Similar to, yet slightly different than kj , we determine the values of hr by obtaining the ratio of the sum of occupancy probabilities of the feasible member nodes of a supernode, for which an arriving nonprioritized call is admitted, multiplied with βi , to the sum of occupancy probabilities of all the feasible member nodes of that particular supernode. In Fig. 3, we show the 1D MC model for non-prioritized calls where each node represents a supernode composed of a set of nodes shown in Fig. 2. The equilibrium channel occupancy probabilities, qnp (r), could be obtained similarly to prioritized calls if unidirectional transition flows, shown in Fig. 2, did not exist. However their existence needs to be taken into account by adjusting μnp affiliated with each supernode appropriately. Therefore we initiate μnp (r) to replace μnp affiliated with each supernode in the model given in Fig. 3 and determine its value by dividing the number of transition flows departing from the associated supernode with the number of pairs of bidirectional transition flows in between the same particular supernodes.
μnp =
(C − r)/bp + 1 · μnp (r), r = 1, . . . , m/bp (9) (m − (r − 1))/bp
Then we can obtain the occupancy probabilities qnp (r), r = 0, 1, . . . , ((m/bnp −1), which satisfy the following recursive equation.
(λnp · hr−1 ) · qnp (r − 1) = r · μnp (r) · qnp (r)
The admission probabilities for prioritized calls, kj , and nonprioritized calls, hr , cannot be obtained without computing the occupancy probability of each feasible node. Even if the occupancy probabilities of supernodes for prioritized and nonprioritized calls can be obtained using this method, we still need to compute the occupancy probabilities of certain feasible nodes since joint occupancy probabilities of these supernodes cannot be used due to their dependencies. To overcome these difficulties, we suggest the following iterative approach: 1) Initialize the values of estimated equilibrium occupancy probabilities (ˆ q (nnp , np ) for nnp = 0, 1, . . . , m/bnp and np = 0, 1, . . . , C/bp ) by setting them to 1/(total number of feasible nodes). 2) Calculate μnp (r) for r = 1, . . . , m/bnp using (9). 3) Iterate with the following steps until the changes in the updated values of kj and hr are less than a chosen resolution. for j = a) Calculate and update kj 0, 1, . . . , (C/bp − 1) and qp (j) for j = 0, 1, . . . , (C/bp ) using (7) and (8). b) Update the values of estimated occupancy probabilities, qˆ(nnp , np), by apportioning the value of the last updated occupancy probability, qp (j), of the corresponding supernode for prioritized calls amongst its nodes with respect to the value of the last updated occupancy probability, qnp (r), of the corresponding supernode for non-prioritized calls. = c) Calculate and update hr for r 0, 1, . . . , (m/bnp − 1) and qnp (r) for r = 0, 1, . . . , (m/bnp ) using (11) and (12). d) Update the values of estimated occupancy probabilities, qˆ(nnp , np ), by apportioning the value of the last updated occupancy probability, qnp (r), of the corresponding supernode for non-prioritized calls amongst its nodes with respect to the value of the last updated occupancy probability, qp (j), of the corresponding supernode for prioritized calls. 4) Obtain call blocking probabilities for prioritized and non-prioritized calls using qˆ(nnp , np ). The call blocking probabilities for both types of calls are calculated as follows when the final estimated values of equilibrium occupancy probabilities, qˆ(nnp , np ), are obtained. m/bnp
Bp =
(10)
r = 1, . . . , m/bnp m/bnp qnp (r) = 1 we Solving for qnp (0) in the equation r=0 obtain
r−1 (λnp ·hz ) z=0 ( μnp (r) ) · qnp (0), 1 ≤ r ≤ m/bnp (11) qnp (r) = r! where ⎡
r−1 λnp ·hn ⎤−1 m/bnp n=0 ( μnp (r) ) ⎦ (12) qnp (0) = ⎣1 + r! r=1
qˆ(n, (C − (n · bnp ))/bp )
(13)
n=0 m/bnp (C−(a·bnp ))/bp
Bnp =
a=0
n=0
qˆ(a, n),
(14)
(a · bnp + n · bp ) ≥ m/bnp Despite its iterative nature, we expect the proposed state-space decomposition method to have a low computational cost since decomposing the whole state space into subspaces and forming supernodes enables applications of 1D MC modeling and corresponding closed form formulas, which makes the proposed method efficient to implement for real time applications.
YAVUZ and LEUNG: EFFICIENT METHODS FOR PERFORMANCE EVALUATIONS OF CALL ADMISSION CONTROL SCHEMES
λ p .k 0
λ p .kC −( ⎣m / bnp ⎦.bnp )−bp λ p .kC −( ⎣m / bnp ⎦.bnp ) λ p .kC −( ⎣C / bp ⎦.bp )−bp
λ p .k1 2b p
bp
0
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(m+1)μ p
⎣C / b p ⎦.b p
C − ( ⎣m / bnp ⎦.bnp )
λnp .β 1 .h1
λ np .β 0 .h0
2bnp
bnp
0
μ'p
λ np .β ⎣m / bnp ⎦.bnp −bnp .h⎣m / bnp ⎦.bnp −bnp
λnp .β 2 .h2
2μ'p
3μ'p
Cμ'p ⎣m / bnp ⎦.bnp
Fig. 3.
One dimensional MC model obtained for the prioritized and non-prioritized calls’ supernodes.
0
IV. N UMERICAL R ESULTS
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Blocking probability of prioritized calls
In this section we compare the performance of the proposed method, state-space decomposition, with Borst and Mitra’s approximation [28] and the direct numerical methods for asymmetric CAC schemes. We then show that the runtime computational cost of the proposed approximation method is negligible compared to existing numerical methods’ (i.e., direct, method of Jacobi, method of Gauss-Seidel) with respect to CPU time and memory needed to obtain the results. We investigate the cutoff priority scheme, which is a special case of the fractional guard channel scheme, using the following set of parameters: C = 32, m = 24, (βi = 1 for i = 0, . . . , 23, and 0 otherwise) λnp = 0.1, 1/μnp = 200, 1/μp = 50, bp = 1 and 3, bnp = 1 and λp is varied from 1 to 0.05. However any fractional guard channel scheme can be chosen since other choices of βi ’s would give similar results. A two-class model is considered for the sake of simplicity and to challenge Borst and Mitra’s assumption on independent channel occupancy. Figs. 4 and 5 depict the prioritized and non-prioritized call blocking probabilities respectively under varying prioritized call traffic load. We observe for both values of bp that, when ρp > ρnp , both call blocking probabilities approximated by the proposed method match the exact results very well. However Borst and Mitra’s method overestimates the prioritized call blocking probabilities generously while it underestimates the non-prioritized ones extensively. When ρp < ρnp , the proposed method slightly overestimates the prioritized call blocking probabilities while it slightly underestimates the nonprioritized ones with the discrepancy increasing as both traffic loads are decreasing. Yet, Borst and Mitra’s method gives a better approximation only when both traffic loads are very low due to its assumption on independent channel occupancy. The discrepancy observed when ρp < ρnp is due to an assumption that we made in the iterative solution described above; i.e., the steady state probabilities of all nodes that are members of the same particular supernode for prioritized calls are proportional to each other with the same ratio that exists between the steady state probabilities of the corresponding supernodes for non-prioritized calls, and vice versa. Therefore we expect the proposed method to approximate steady state
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Direct method, b = b = 1 p np Borst−Mitra method, b = b = 1 p np Proposed method, b = b = 1 p np Direct method, b = 3, b = 1 p np Borst−Mitra method, b = 3, b = 1 p np Proposed method, b = 3, b = 1
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Fig. 4. Prioritized call blocking probability for the cutoff priority scheme bp = 1 and 3, bnp = 1, and m = 24.
probabilities of nodes that are members of supernodes which have a relatively less number of member nodes better with respect to others that are members of supernodes which have relatively more number of member nodes. However this is not a significant problem unless ρnp >> ρp , since call blockings mostly occur at nodes that are members of supernodes which have relatively less number of member nodes and thus closer to the edges of the respective transition diagrams. When ρnp > ρp , steady state probabilities of the nodes that have a relatively more number of non-prioritized calls dominate the others and thus lead to the discrepancy. On the other hand, Borst and Mitra’s method gives better approximations only when traffic loads of both classes are very low due to the channel occupancy independence assumption [28]. When each of the individual classes accounts for a substantial portion of the total amount of capacity in use, it leads to mutual dependence as the traffic load increases. Both approximation methods perform relatively better when the capacity requirement for prioritized calls, bp , increases. When the number of shared channels, m, is increased to
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Blocking probability of non−prioritized calls
28, the proposed method gives more accurate estimates of call blocking probabilities for both types of calls since the transition diagram has more supernodes for non-prioritized calls that have a relatively less number of member nodes. We make another comparison in Figs. 6 and 7 by using the following set of parameters: C = 32, m = 28, λnp = 0.1, 1/μnp = 200, 1/μp = 50, bnp = 1 and 3, bp = 1 and λp is varied from 1 to 0.05. The results are similar to the first case; however, when bnp increases, the non-prioritized call blocking probabilities become less accurate for the proposed method but more accurate for Borst and Mitra’s method. Yet, Borst and Mitra’s method still yields a better approximation than the proposed method only when the loads of both traffic classes are very low. In [20] we proposed a closed form approximation method, effective duration time, to evaluate call blocking performance in cellular networks under homogeneous traffic. The method provides accurate results, yet the results are sensitive to the average values of channel holding times. We compare the performance of this method to the proposed state-space decomposition method’s to observe if it is more sensitive than the previously proposed method under homogeneous traffic. When call traffic load is varied by changing the value of call arrival rates we observe that both methods slightly overestimate the prioritized call blocking probability when ρp < ρnp whereas the results obtained from both approximation methods match the results obtained from the direct numerical method very well when ρp > ρnp . The state-space decomposition method underestimates the non-prioritized call blocking probability while the effective duration time method provides results that match well with the exact solutions. On the other hand, when call traffic load is varied by changing the value of average channel holding times we observe that the results obtained from both approximation methods match the ones obtained by the direct numerical method very well when ρp > ρnp , whereas the effective duration time method degenerates slightly compared to the previous results given above when ρp < ρnp . We observe that the proposed statespace decomposition method is indifferent to changes in average channel holding times as opposed to the effective duration time method. For real time applications, computationally efficient approximation methods for evaluating CAC schemes are needed to replace such methods as direct which provide exact solutions by solving large sets of flow equations. Product form solutions are preferable due to their computational efficiency; however, it is very difficult to find one to evaluate asymmetric CAC schemes. Considering that the state-space decomposition method is iterative, we need to compare it with the direct and other widely used iterative methods such as the Jacobi method, Gauss-Seidel method and approximate method such as BorstMitra with respect to their computational costs to evaluate its benefits. We compare the runtime computational costs of the numerical and the approximation methods using the parameters ”CPU time” and ”used memory”. We are aware that the results depend on the computer used; however our intent is only to give the readers an idea on the relative performance of the different methods rather than to provide absolute performance figures. We define ”CPU time” as the total processing time (in
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Fig. 6. Prioritized call blocking probability for the cutoff priority scheme bp = 1, bnp = 1 and 3, and m = 28.
seconds) consumed for the computation using each method and the ”used memory” as the amount of storage allocated for nonzero matrix elements. We use the following set of parameters to obtain the numerical results presented in Table II by evaluating the cutoff priority asymmetric CAC scheme for three different scenarios: C = 6, m = 5, λnp = 0.02; C = 32, m = 28, λnp = 0.1; C = 64, m = 56, λnp = 0.5, while 1/μnp = 200, 1/μp = 50, bp = bnp = 1 and λp varies from 0.2 to 0.01, 1 to 0.05 and 5 to 0.25, respectively. The parameters are chosen to obtain similar values of call blocking probabilities for both classes of calls in all three scenarios to make the comparisons appropriate. The results given in Table II show that the CPU times and the used memory obtained from our approximation method is almost negligible when compared with the ones obtained from the direct numerical solution, Jacobi method and Gauss-Seidel method especially when the number of channels in the system increases. Decomposing the whole state space into subspaces and forming supernodes to enable iterative applications of
YAVUZ and LEUNG: EFFICIENT METHODS FOR PERFORMANCE EVALUATIONS OF CALL ADMISSION CONTROL SCHEMES
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Seidel. We believe that the easy-to-implement approximation method with low computational costs for performance evaluations, as proposed in this paper, will help motivate the practical application of dynamic CAC schemes that adaptively adjust resources allocated to multiple service classes based on computation of CAC performance metrics in real time.
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Fig. 7. Non-prioritized call blocking probability for the cutoff priority scheme bp = 1, bnp = 1 and 3, and m = 28.
1D MC modeling with its closed form formulas make our proposed approximation method’s computational cost comparable with that of solutions obtained from single closed form formulas in terms of CPU time and memory usage. This is because our evaluations show that the admission probabilities for the prioritized calls, kj , and the non-prioritized calls, hr , converge very fast. Therefore both the methods of state-space decomposition and Borst-Mitra perform similar to a closed form formula solution with respect to CPU times and used memory given in Table II. The latter computes the solution faster using less memory compared to the former; however the proposed method gives more accurate approximations under a wide range of conditions with only a relatively small increase in computational cost. V. C ONCLUSION In this paper we have classified CAC schemes according to their MC models into two categories: symmetric and asymmetric. We have obtained a product-form exact solution to evaluate symmetric CAC schemes in multi-service networks. To evaluate the call blocking performance of asymmetric CAC schemes in multi-service networks, for which closed form solutions are difficult to obtain, we have proposed a novel approximation method called state-space decomposition that uses an iterative approach and is therefore computationally efficient. We have compared the numerical results obtained from the proposed state-space decomposition method, with those obtained from a previously proposed approximation method by Borst and Mitra, and the direct numerical method that provides the exact solution. The results presented show that the proposed method provides a better match to the exact solutions compared to the Borst-Mitra method over a wide range of conditions. We have evaluated the computational costs of various schemes. Results show that the CPU time and memory usage for the proposed method are slightly higher than those of the Borst-Mitra method, and both these methods have an almost negligible computation cost compared with exact solution methods such as direct, Jacobi and Gauss-
[1] L. Huang, S. Kumar, and C. C. J. Kuo, “Adaptive resource allocation for multimedia services in wireless communication networks,” in Proc. 21st International Conference on Distributed Computing Systems Workshop (ICDCSW ’01), pp. 307–312, 2001. [2] D. Hong and S. S. Rappaport, “Traffic model and performance analysis for cellular mobile radiotelephone systems with prioritized and nonprioritized handoff procedures,” IEEE Trans. Veh. Technol., vol. 35, pp. 77–92, Aug. 1986. [3] B. Li, C. Lin, and S. T. Chanson, “Analysis of a hybrid cutoff priority scheme for multiple classes of traffic in multimedia wireless networks,” Wireless Networks, vol. 4, no. 4, pp. 279–290, July 1998. [4] Y. B. Lin, S. Mohan, and A. Noerpel, “Queuing priority channel assignment strategies for handoff and initial access for a PCS network,” IEEE Trans. Veh. Technol., vol. 43, no. 3, pp. 704–712, Aug. 1994. [5] J. Y. Lee and S. Bahk, “Simple admission control schemes supporting QoS in wireless multimedia networks,” IEEE Electron. Lett., vol. 37, no. 11, pp. 712–713, May 2001. [6] R. Ramjee, R. Nagarajan, and D. Towsley, “On optimal call admission control in cellular networks,” Wireless Networks, vol. 3, no. 1, pp. 29– 41, Mar. 1997. [7] Y. Fang and Y. Zhang, “Call admission control schemes and performance analysis in wireless mobile networks,” IEEE Trans. Veh. Technol., vol. 51, no. 2, pp. 371–382, Mar. 2002. [8] M. D. Kulavaratharasah and A. H. Aghvami, “Teletraffic performance evaluation of microcell personal communication networks (PCN’s) with prioritized handoff procedures,” IEEE Trans. Veh. Technol., vol. 48, no. 1, pp. 137–152, Jan. 1999. [9] A. S. Acampora and M. Naghshineh, “Control and quality of service provisioning in high-speed micro-cellular networks,” IEEE Personal Commun., vol. 1, no. 2, pp. 36–43, 1996. [10] M. Naghshineh and S. Schwartz, “Distributed call admission control in mobile/wireless networks,” IEEE J. Select. Areas Commun., vol. 14, no. 4, pp. 711–717, May 1996. [11] D. Levine, I. Akyildiz, and M. Naghshineh, “A resource estimation and call admission algorithm for wireless multimedia networks using the shadow cluster concept,” IEEE/ACM Trans. Networking, vol. 5, no. 1, pp. 1–12, Feb. 1997. [12] A. Sutivong and J. M. Peha, “Novel heuristics for call admission control in cellular systems,” in Proc. IEEE 6th International Conference on Universal Personal Communications Record, vol. 1, pp. 129–133, Oct. 1997. [13] C. Oliveira, J. B. Kim, and T. Suda, “An adaptive bandwidth reservation scheme for high-speed multimedia wireless networks,” IEEE J. Select. Areas Commun., vol. 16, pp. 858–874, Aug. 1998. [14] P. Ramanathan, K. M. Sivalingam, P. Agrawal, and S. Kishore, “Dynamic resource allocation schemes during handoff for mobile multimedia wireless networks,” IEEE J. Select. Areas Commun., vol. 17, no. 7, pp. 1270–1283, July 1999. [15] R. Guerin, “Equivalent capacity and its application to bandwidth allocation in high-speed networks,” IEEE J. Select. Areas Commun., vol. 9, no. 7, pp. 968–981, Sept. 1991. [16] J. S. Evans and D. Everitt, “Effective bandwidth-based admission control for multiservice CDMA cellular networks,” IEEE Trans. Veh. Technol., vol. 48, no. 1, pp. 36–46, Jan. 1999. [17] Q. Ren and G. Ramamurthy, “A real-time dynamic connection admission controller based on traffic modeling, measurement, and fuzzy logic control,” IEEE J. Select. Areas Commun., vol. 18, no. 2, pp. 184–196, Feb. 2000. [18] Y. Fang and I. Chlamtac, “Teletraffic analysis and mobility modeling for PCS networks,” IEEE Trans. Commun., vol. 47, pp. 1062–1072, July 1999. [19] Y. Fang, I. Chlamtac, and Y. B. Lin, “Channel occupancy times and handoff rate for mobile computing and PCS networks,” IEEE Trans. Comput., vol. 47, pp. 679–692, June 1998. [20] E. A. Yavuz and V. C. M. Leung, “Computationally efficient method to evaluate the performance of guard-channel-based call admission control in cellular networks,” IEEE Trans. Veh. Technol., vol. 55, no. 4, pp. 1412–1424, July 2006.
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TABLE II C OMPARISON OF COMPUTATION COMPLEXITY BETWEEN THE PROPOSED APPROXIMATION METHOD AND THE Borst-Mitra, direct AND iterative METHODS Total number of channels = C, total number of shared channels = m C = 6, m = 5
C = 32, m = 28
C = 64, m = 56
Numerical Methods
CPU time
used memory
CPU time
used memory
CPU time
used memory
Direct Method Jacobi Method Gauss-Seidel Method Borst - Mitra Method Proposed method
0.07s 0.25s 0.18s 0.003s 0.006s
3.82e+03 2.46e+03 2.46e+03 0.12e+03 0.13e+03
30.74s 29.96s 1m46s 0.02s 0.04s
152e+04 91.6e+04 91.6e+04 0.05e+04 0.14e+04
49m8s 11m24s 1h33m24s 0.033s 0.45s
222.52e+05 133.64e+05 133.64e+05 0.011e+05 0.046e+05
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Emre Altug Yavuz (S’94) received the B.Sc. and M.A.Sc. degrees in electrical and electronics engineering from the Middle East Technical University (METU) in Ankara, Turkey, in 1995 and 1998, respectively. From 1999 to 2001, he was a Software Engineer with ALCATEL in Toronto, Canada, developing safety critical real time microprocessor firmware for embedded command, control, and communication applications in automated train systems. He received the Ph.D. degree in electrical and computer engineering from the University of British Columbia (UBC) in Vancouver, Canada in January 2007. He is currently working as a consultant in Telecom industry in Vancouver. His research interests include performance analysis, quality of service, admission and congestion control, cross-layer design, and traffic modeling in wireless networks. Victor C. M. Leung (S’75-M’89-SM’97-F’03) received the B.A.Sc. (Hons.) degree in electrical engineering from the University of British Columbia (U.B.C.) in 1977, and was awarded the APEBC Gold Medal as the head of the graduating class in the Faculty of Applied Science. He attended graduate school at U.B.C. on a Natural Sciences and Engineering Research Council Postgraduate Scholarship and obtained the Ph.D. degree in electrical engineering in 1981. From 1981 to 1987, Dr. Leung was a Senior Member of Technical Staff and satellite systems specialist at MPR Teltech Ltd., Canada. In 1988, he was a Lecturer in Electronics at the Chinese University of Hong Kong. He returned U.B.C. as a faculty member in 1989, where he is a Professor and the holder of the TELUS Mobility Research Chair in Advanced Telecommunications Engineering in the Department of Electrical and Computer Engineering. His research interests are in wireless networks and mobile systems. Dr. Leung is a Fellow of IEEE and a member of ACM. He serves on the editorial boards of the IEEE T RANSACTIONS ON W IRELESS C OMMU NICATIONS , IEEE T RANSACTIONS ON V EHICULAR T ECHNOLOGY , IEEE T RANSACTIONS ON C OMPUTERS , and the I NTERNATIONAL J OURNAL OF S ENSOR N ETWORKS . He has served on the Technical Program Committees (TPC) of numerous conferences, and was the TPC Vice-Chair of IEEE WCNC 2005. He was also the General Co-chair of ACM/IEEE MSWiM 2005, and the General Chair of Qshine 2007.
