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A Dynamic Call Admission Policy with Precision QoS Guarantee Using Stochastic Control for Mobile Wireless Networks  Si Wuy, K. Y. Michael Wong

Department of Physics Hong Kong University of Science and Technology Clear Water Bay, Kowloon Hong Kong E-mail: [email protected] [email protected]

Bo Li

Department of Computer Science Hong Kong University of Science and Technology Clear Water Bay, Kowloon Hong Kong E-mail: [email protected] Corresponding author: K. Y. Michael Wong Tel: +852 2358-7480 Fax: +852 2358-1652 submit to IEEE/ACM Transactions on Networking

 This work is supported in part by grants HKTIIT92/93.002 and HKTIIT97/98.EG01 from Hong Kong Telecom

Institute of Information Technology (HKTIIT). Preliminary results were presented in [16]. y Present address: Lab. for Information Synthesis, RIKEN Brain Science Institute, Hirosawa 2-1, Wako-shi, Saitama 351-01, Japan.

1

Abstract Call admission control is one of the key elements in ensuring the QoS in mobile wireless networks supporting multimedia applications. The traditional trunk reservation policy and its numerous variants give preferential treatment to the hando calls over new arrivals by reserving a number of radio channels exclusively for handos. Such schemes, however, cannot adapt to changes in trac pattern due to the static nature. This paper introduces a novel stable dynamic call admission control mechanism (SDCA), which can maximize the radio channel utilization subject to a predetermined bound on the call dropping probability. The novelties of the proposed mechanism are: 1) it is adaptive to wide range of system parameters and trac condition due to its dynamic nature; 2) the model takes into account the eects of limited channel capacity and time dependence on the call dropping probability, and the in uences from nearest and next-nearest neighboring cells, which greatly improve the control precision; 3) the control mechanism is based on a probabilistic admission policy (acceptance ratio) can spread new arrivals evenly over a control period, which leads to more eective and stable control. Most of the computational complexities are pre-calculated o-line except for solving the non-linear equation to obtain the acceptance ratio, in which a coarse-grain numerical integration is shown to be sucient for stochastic control. Extensive simulation results show that our scheme steadily satis es the hard constraint on call dropping probability while maintaining a high channel throughput.

1 Introduction There has been a rapid development in wireless cellular communications. The next generation of networks are expected to eventually carry multimedia trac - voice, video, images, or data, or combinations of them. Various issues related to this future wireless mobile networks carrying multimedia trac have to be carefully examined before such systems can be realized [15]. These include issues ranging from the physical layer, channel access protocol, channel allocation, hando control and etc. The Quality of Service (QoS) guarantee is essential in networks supporting multimedia. One of the key elements in providing QoS guarantee is an eective Call Admission Control (CAC) policy, which not only has to ensure that the network meets the QoS of the newly arriving calls if accepted, but also guarantees that the QoS of the existing calls does not deteriorate. This paper deals with admission control related to the radio channel assignment. When a mobile moves across cells during its lifetime, dropping is primarily caused by the unavailability of the channels in the new cell. Dropping a call in progress is generally considered to have more negative impact from users' perception than rejecting (blocking) a newly requested call. Therefore, one of the key design goals is to minimize the call dropping probability, which is precisely the objectives of most existing proposals on call admission control. This, however, usually comes at the expense of potentially poor channel utilization by admitting less new calls. Given that radio channels are considered to be the primary scarce resource in mobile wireless networks, the main challenge in the design of an ecient admission control scheme is to balance these two con icting requirements. Hence the major performance parameters of interest in this paper are the call dropping probability, channel utilization and new call blocking probability. There are a number of unique aspects in the next generation of multimedia wireless networks that the design of an eective admission control scheme needs to take into account: 2

 Smaller cells will be employed (micro-cells or pico-cells), thus the number of handos during

a call's lifetime is likely to be increased; additionally, there is an increased in uence from neighboring cells and even next neighboring cells [14].  Possibly dierent QoS requirements for dierent calls, and potentially more stringent QoS requirements of individual calls mandate a highly precise resource allocation [8].  Diversi ed trac load requires that admission control has to be adaptive to the changing trac pattern. Therefore, a dynamic approach is preferred.

This paper introduces a novel stable dynamic call admission control mechanism (SDCA) that aims to maximize the radio channel utilization while satisfying a pre-determined bound on the call dropping probability The novelties of the proposed mechanism are: 1) It is dynamically adaptive to a wide range of system parameters and trac condition due to its dynamic nature; 2) The model takes into account the eects of limited channel capacity and time dependence on the call dropping probability, and the in uences from nearest and next-nearest neighboring cells, which greatly improve the control precision; 3) The control mechanism is based on a probabilistic admission policy (acceptance ratio) can spread new arrivals evenly over a control period, which leads to more effective and stable control. We compare our method with a static control, and a recently-proposed approximate method for dynamic control. Extensive simulation results show that our scheme outperforms the others by steadily satisfying the hard constraint on call dropping probability while maintaining a high channel throughput. The paper is organized as follows. We review the relevant work in Section 2. In Section 3, we present the control algorithm. In Section 4 we study the performance of the model, and compare the results with other proposals. Section 5 concludes the paper. Mathematical details for deriving the algorithm are presented in Appendices A to C.

2 Call Admission Strategies The rationale behind the traditional guard channel scheme (or trunk reservation policy) is to give preferential treatment to the hando calls, which reserves a xed number of channels exclusively for handos [4, 11]. This scheme was shown by Ramjee et al [13] to be optimal for the linear objective functions of the dropping and blocking probabilities de ned above. However, it has a number of de ciencies, in particular, the guard channel scheme cannot satisfy the hard constraints on the call dropping probability often required by multimedia applications. The fractional guard channel policy proposed in [13] is shown to be optimal for minimizing the call blocking probability subject to a hard constraint on the call dropping probability. In addition, there has been numerous extensions based on the guard channel scheme. Epstein and Schwartz considered a mixed trac type with narrow-band and wide-band calls [2]. Another proposal by Acampora and Naghshineh suggests to cluster a group of neighboring cells and allocates a portion of the channels from those cells for handos [1]. Li et al extended the guard channel scheme to handle multiple streams of trac, each having potentially dierent QoS requirements, thus requiring potentially dierent channel thresholds [8]. All such policies, however, are static in that they cannot adapt to changes in the trac pattern. A number of recent proposals have made ne attempts to implement dynamic control in the 3

above schemes. The proposed schemes make the admission decision in a distributed manner relying on the status information exchange between adjacent cells, taking into consideration the active calls in the cell where a new call arrives as well as its neighboring cells where the call is likely to be handed o to. The Shadow Cluster mechanism by Levine et al [7] is based on the observation that \every mobile terminal with an active wireless connection exerts an in uence upon the cells (and their base stations) in the vicinity of its current location and along its direction of the travel". The coverage of a shadow cluster for a given active mobile mainly consists of the cell where the mobile is currently present (i.e., the center of the shadow cluster) and all its adjacent cells along the direction of the travel. This area changes when the mobile call is handed o to other cells, thus a tentative shadow cluster needs to be implemented for every new call as well as every hando call. Simulations show that the shadow cluster mechanism is able to reduce the percentage of dropped calls in a controlled fashion. The eciency of this scheme, however, depends on the accuracy of prediction of the future mobile movement, which makes it most suitable for strong directional environment such as highway. One practical drawback of this scheme is the potentially high computation and communication complexities in re-calculating the shadow cluster for each new call as well as each hando. The computational complexity is reduced in the adaptive admission control scheme by Misic it al [9]. Resource estimations are triggered by the events of call hando, origination and termination. Bandwidths reserved for possible handos are determined by computing the "spatial activity factors". These are time-averaged estimates of the transition probabilities among cells during a call's lifetime, neglecting the dynamical nature of the transitions. As a result, the enforcement of the QoS guarantees is only approximate. In addition, each event, i.e., call handos, origination and termination, requires the update of the status of the admission control policy, which makes it next to impossible to be implemented. The Distributed Call Admission (DCA) scheme by Naghshineh and Schwartz [10], on the other hand, does not need the status information exchange upon each call arrival (new call and hando). Rather, it only requires the exchange of such information periodically [10]. The admission control algorithm calculates the maximum number of calls that can be admitted to a given cell without violating the QoS of the existing calls in the cell as well as its adjacent cells. One of the main features of the DCA is its simplicity in that the admission decision can be made in real time and does not require much computational eort. In addition, the DCA mechanism does not require the directional environment that shadow cluster needs, which makes it more suitable for urban areas. The DCA mechanism is based on a number of simplifying approximations, which lead to imprecise control decisions. (a) It approximates the dropping probability by the tail of a Gaussian distribution, which is applicable only for cells with in nite capacity. (b) It neglects the time dependence of the dropping probability, and the estimate at the end of the control period is assumed to be the average. (c) It approximates that all the admitted new calls are in progress at the beginning of the control period. (d) It neglects the probability that a call can hand o more than once. As a result, the performance of DCA becomes very sensitive to network load. Generally speaking, it yields an excessively low dropping probability at intermediate trac, which grows rapidly with increasing trac load.

4

3 The Control Algorithm We consider a cellular network consisting of close packed hexagonal cells and using a xed channel allocation scheme. Each cell has a capacity of N channels. New calls arrive in cell i at a rate of i . Connected calls terminate at a rate of  (i.e., 1= is the average call duration time). In addition, P calls hand o from cell k to a neighboring cell i at a rate of hik . Let hk  i hik be the hando rate out of cell k. The objective of our call admission scheme is to maximize the channel utilization (minimize the new call blocking probability) subject to a hard constraint that the hando dropping probability should be maintained below a prede ned threshold PQoS required by a QoS guarantee. In a dynamic and distributed control scheme, this is implemented by periodically exchanging status information between neighboring, and even next neighboring cells if necessary. Each cell updates its control action at the beginning of the control period of duration T . The exchanged information includes the channel occupancies and the new call arrival rates. In SDCA, the control action is to determine for the next control period the fraction of new calls to be admitted. We summarize the key features of the algorithm below: (a) We estimate the time-dependent dropping probability in a cell, taking into account its nite capacity. The derivation is based on the solution to the evolution equation of the occupancy distribution. It greatly improves over the Gaussian approximation. (b) We compute the average dropping probability over a control period, taking into account its time dependence. This increases the precision over a single-value approximation within the control period. (c) To alleviate the eects of multiple handos over a control period, we base our estimation of the dropping probability on the call transition probabilities between nearest as well as 2nd and 3rd nearest neighboring cells. We show that this has signi cant impact under certain parameters. While the exact computation of the transition probabilities involve the exponentiation of a matrix whose dimension is the number of cells in the network, we introduce a local approximation which reduces the computational complexity, yet yields an excellent precision. (d) The QoS requirement on the dropping probability yields an expression for the acceptance ratio ai , which is the maximum fraction of new calls to be admitted into cell i in the coming control period. Instead of using it to determine the admission threshold as in a guard channel policy, we spread the new calls uniformly over the period, by stochastically accepting each new call with probability ai . This avoids a sudden overload of the network at the beginning of the control period during congestions, leading to more eective and stable control. (e) The acceptance ratio is obtained by solving a non-linear equation for the average dropping probability on-line. This constitutes the major computational complexity of the algorithm. However, since the control is stochastic, a coarse-grain integration of the average dropping probability is already sucient. On the other hand, the call transition probabilities can be pre-calculated o-line for constant hando rates, either by exact matrix computation or local approximation. The rest of the section describes the details of the computation. The key is to derive the acceptance ratio ai for cell i periodically, which is obtained via Eq. (21) according to the three steps. (a) We compute the inter-cellular transition probabilities using a local approximation, and 5

hence the mean and variance of the time-dependent occupancy distribution in each cell, summarized in Eqs. (8) and (9). (b) We derive a diusion equation whose solution describes the evolution of the occupancy distribution. (c) We introduce a mean-rate approximation which enables us to obtain the dropping probability in Eq. (19) by combining the results of (a) and (b).

3.1 The Local Approximation Consider the single-call transition probability fik (t) that an ongoing call in cell k at the beginning of the control period (t = 0) is located in cell i at time t. For an eective control enforcing dropping probabilities of the order 10?3 to 10?2 , we assume that essentially all calls hand o successfully, resulting in the evolution equation

dfik (t) = ? X J f (t) and f (0) =  ; ij jk ik ik dt j

(1)

where Jik is the transition matrix given by Jii = hi +  and Jik = ?hik for i 6= k. The solution to (1) is fik (t) = [exp(?Jt)]ik : (2) P ?1 , where  is The matrix elements could be obtained by using [exp(?Jt)]ik =  Ui e? t Uj an eigenvalue of J , and Ui is the ith element of the corresponding eigenvector. However, the computational complexity of this matrix operation can be reduced by considering the o-diagonal terms as perturbations to the diagonal part of J . Each term in the resultant perturbation series of fik (t) corresponds to the contribution of a path connecting k and i by cell hopping. For illustration, we consider in Appendix A the case of homogeneous hando rates, i.e. hik = h=6 for all pairs of nearest neighbors i and k. In this case the contribution of a path takes the simple form  n ht 1 qn (t) = n! 6 exp[?(h + )t];

(3)

where n is the number of hops along the path from k to i. Hence fik (t) is obtained by summing over all possible paths between k and i. For the cellular network in Fig. 1(a), Fig. 1(b) shows the example of k = i, in which each diagram represents the topology of a path connecting i to itself, with vertices and edges representing cells and paths respectively. Hence there are one path of 0 hop, no path of 1 hop, six paths of 2 hops and twelve paths of 3 hops, leading to fik (t) = q0(t) + 6q2 (t) + 12q3 (t) +


: (4) Similarly, Fig. 1(c) shows the paths connecting k; i = nearest neighbors, giving fik (t) = q1(t) + 2q2 (t) + 15q3 (t) +


: (5) Likewise,

fik (t) = 2q2 (t) + 6q3 (t) +


; fik (t) = q2 (t) + 6q3 (t) +


; 6

for k; i = 2nd nearest neighbors; for k; i = 3rd nearest neighbors:

(6)

Since ht is the average number of hops in time t, the resultant perturbation series is rapidly converging for ht up to O(1). For a hando rate h as high as 0.05 s?1 and t = 20 s,  = 0.005 s?1 , the computed values for fii (t) are lower than the true values by 1% up to 2 hops, and 0.3% up to 3 hops. The transition probabilities enable us to estimate the distribution of ongoing calls in cell i at time t. At t = 0 there are nk0 calls in cell k initially. Since all events (call arrivals, handos and terminations) are stochastic, the number ofPongoing calls in cell i at time t is then a superposition P of binomial variables, with resultant mean k fik (t)nk0 and variance k fik (t)[1 ? fik (t)]nk0 . For the new calls arriving in cell k at time t0 , the probability of nding them in cell i at time t is fik (t ? t0 ). Since they are evenly distributed over time, the number of new calls in cell i at time t obeys a Poisson distribution with mean Pk gik (t)ak k , where gik (t) is the integrated transition probability given by

gik (t) =

Z

0

t 0 dt f

0

ik (t ? t ):

(7)

Knowing the algebraic expressions of fik (t), it is straightforward to obtain close forms for gik (t). Thus the mean of the occupancy distribution pni (t) in cell i at time t is given by

hni(t)i = and the variance is

i(t)2 =

X

k

X

k

fik (t)nk0 +

X

k

gik (t)ak k ;

fik (t)[1 ? fik (t)]nk0 +

X

k

gik (t)ak k :

(8)

(9)

Evidently, (8) and (9) are applicable to long control periods, and the label k can include cells up to arbitrary distance. In practice, we truncate the summation beyond nearest and next-nearest neighboring cells, since long-range cell hopping can be neglected in a single control period. The transition probabilities for inhomogeneous hando rates are derived in Appendix B.

3.2 The Diusion Equation As shown in Fig. 2(a), the initial channel occupancy distribution at the beginning of a control period is a delta function given by pni (0) = ni ;ni0 . At a subsequent time t within the control period, the distribution broadens because of the stochastic nature of the events of call handos, arrivals and departures, as schematically shown in Fig. 2(b). For large system size, the distribution evolves into a Gaussian distribution with mean hni (t)i and variance i (t)2 . However, the limited capacity of cells modi es the occupancy distribution to non-Gaussian, especially for nearly full occupancy, which is the region of interest in estimating the dropping 7

probability (Fig. 2(c)). The modi ed distribution is derived by considering the evolution equation for pni (t):

dpn(t) = p (t) ? ( + M)p (t) + Mp (t); n < N; n?1 n n+1 dt

(10)

dpn(t) = p (t) ? Mp (t); n = N; n?1 n dt

(11)

where  and M are the total arrival and departure rates for calls in cell i. Their dependence on n and t are assumed to be negligible. (Unless explicitly speci ed, the subscript i is omitted hereafter.) In the limit of large N , the evolution equation (10) reduces to a diusion equation for the continuous distribution P (x; t), where x  n=N :

@P (x; t) = ?v @P (x; t) + D @ 2 P (x; t) ; @t @x @x2

(12)

where v  ( ? M)=N is the drift velocity, and D  ( + M)=2N 2 is the diusion coecient, in analogy with particle diusion [6]. To consider the boundary conditions in the limit of large N , we rst estimate the scaling of various terms. In typical network problems,   M  O(N ). For an eective admission control, the arrival rate  should be adjusted so that  ? M vanishespto the leading order O(N ),pand only statistical uctuations should contribute, hence  ? M  O( N ). This yields v  O(1= N )pand D  O(1=N ), and (12) implies that P (x; t) should vary signi cantly over a range of x  O(1= N ) and a range of t  O(1). Applying the scaling argument to (11), the time derivative of pn (t) becomes negligible, and we arrive at the boundary condition on the right hand side (x; t) at x = 1; vP (x; t) = D @P@x

p

(13)

Since P (x; t) varies signi cantly in the range x  1 ? O(1= N ), we assume that the boundary condition on the left hand side is P (x; t) = 0 at x = ?1. The initial condition is P (x; t) = (x ? x0 ); at t = 0; (14) where x0 = ni0 =N .

8

This equation is solved in Appendix C. In particular, the solution at the boundary x = 1 is 2i

  exp ? (1?x40Dt?vt) p P (1; t) = 2 + Dv H 1 ?px0 ? vt ; 4Dt 2Dt h

(15)

where

H (x) =

Z

1

x

du exp(p?u =2) ; 2 2

(16)

p

which is related to the complementary error function via H (x) = erfc(x= 2)=2 [12]. The dropping probability is given by D(t) = pN (t) = P (1; t)=N .

3.3 The Mean-Rate Approximation By (15), the dropping probability is determined by the drift velocity v and the diusion coecient D. They determine, respectively, the evolution of the peak and width of the occupancy distribution. To estimate these parameters, we focus on the distribution for occupancies far away from the boundary x = 1. For an eective control, this is the region with dominant contribution to the distribution, and hence should give a reliable estimate on the peak and width. In this case, the boundary condition (13) is replaced by P (x; t) = 0 at x = 1, leading to 2i

0 ?vt) exp ? (x?x4Dt p ; P (x; t) = 4Dt h

(17)

which is a Gaussian distribution with mean x0 + vt and variance 2Dt. This solution also applies to cells with nite capacity except for the region near the boundary. A comparison of (17) with (8) and (9) allows us to identify for cell i, )i ? ni0 ; D = i (t) : v = hni(tNt 2N 2 t 2

(18)

This amounts to a mean-rate approximation, which assumes that the occupancy distribution at time t is the consequence of a constant drift velocity and diusion coecient from time 0 up to t. These constants are assigned so that they yield the correct mean and variance of the occupancy distribution at that particular instant. They are updated for every instant the dropping probability is computed. However, since they are only mildly dependent on time, the approximation is very satisfactory. 9

Hence the dropping probability Di (t) for cell i can be expressed in terms of the quantities ni0 ,

hni(t)i and i(t)2 :

?i(t)2 =2] + 2 [hni (t)i ? ni0] H ( (t)) ; p Di (t) = 2 exp[ i i(t)2 2i (t)2

(19)

where i (t)  (N ?hni (t)i)=i (t) is the normalized vacancy in cell i at time t. The average dropping probability over a control period is obtained by Z T 1 ~ dt Di (t): Di =

T

0

(20)

For stochastic control, the precision for integration needs not be high. We found that it is sucient to use a 7-point Simpson rule [12]. The acceptance ratio ai can be obtained by solving numerically D~ i = PQoS : (21) At low trac, it may happen that D~ i < PQoS even for ai = 1. Then ai is set to 1. Similarly, at high trac, ai is set to 0 if D~ i > PQoS even for ai = 0.

4 Results At the beginning of the control period, cells exchange their status information. To avoid excessive status information exchange, the exchange is limited to 1st, 2nd and 3rd nearest neighboring cells, and the computation of the average dropping probability is truncated accordingly. The information transmitted by cell i includes its cell occupancy ni0 at that instant and the number of admitted new calls i in the previous period. The average new call arrival rate is computed by the moving average (1 ? )ai i + i =T ! ai i . The transition probabilities are computed in the local approximation for paths up to 3 hops. These parameters are then used to compute the admission ratio in cell i, assuming that the admission ratios in other cells take their average values. This is done by substituting (8), (9), (19) and (20) into (21), and solving for ai numerically. We use the bisection method to nd the solution. Simulations were performed on a hexagonal cluster of 19 cells. To alleviate nite size eects we implement periodic connections on the 3 pairs of opposite sides of the cluster. The parameters are N = 100,  = 0:005 s?1 , hi = h = 0:01 s?1 , T = 20 s, and PQoS = 0:01. The dropping probability and new call blocking probability are compared for SDCA, DCA and the trunk reservation scheme (denoted by TR). Since TR is designed for static trac pattern, the dropping probability increases rapidly with the network load when the guard channels are few, but remains too low when the guard channels are many. Here we choose the number of guard channels to be 5 for comparable performance when the network starts to get overloaded. In each case, simulations were done by averaging over 10 samples, each for 1 hour of trac. Below we present the results for a number of cases: 10

(a) Uniform trac: Fig. 3 shows that the proposed method maintains an almost constant dropping probability for a large range of call rates. Results for DCA are qualitatively the same as in [10], except that here the dropping probabilities are presented in linear rather than logarithmic scale. We remark that in most previous work, QoS guarantees were at most achieved up to the same order of magnitude, i.e. the logarithm of the dropping probabilities agrees with the QoS guarantee, whereas here we have enforced the QoS guarantee with a much higher precision. This precision and stability makes it a suitable method for dynamic control. In contrast, both TR and DCA cannot enforce the QoS guarantee for the dropping probability. For TR it grows with the call arrival rate. For DCA it is far below the QoS requirement for call arrival rates up to 1 per second, and Fig. 4 shows that this is accompanied by a substantial sacri ce in the new call blocking probability. For higher call rates, the dropping probability for DCA grows rapidly. Fig. 5 shows that the channel utilization of SDCA is comparable to TR, and signi cantly higher than DCA. (b) Changes in hando rates: Fig. 6 compares the performance of SDCA and TR when the hando rate h changes. At h = 0.01 s?1 , TR with 6 guard channels and SDCA have comparable dropping probabilities over the range of trac being studied. However, when h changes to 0.02 s?1 , TR with the same number of guard channels yields a signi cantly higher dropping probability. On the other hand, SDCA still maintains the same level of dropping probability. This experiment illustrates the adaptability of SDCA. It is true that TR is proved to be optimal [13]. However, when the network situation changes, the parameters for optimal operation also shift. Yet SDCA adapts to the change with a single algorithm. (c) Non-uniform trac: We also simulate a situation for non-uniform trac, in which the central cell has 2 times the normal call rate and its nearest neighbors have 1.5 times the normal call rate and h = 0.05 s?1 . Fig. 7(a) shows that the proposed method continues to maintain a stable level of dropping probability. (d) Inhomogeneous hando rates: We consider the situation in which the hando pattern is inhomogeneous but radially symmetric. Referring to the network in Fig. 1(a), cells are arranged in three consecutive rings. For the innermost ring, h1j = hj 1 = 0:02=6 s?1 . For outer rings, hij = hji and the inward, sideways and outward hando rates have the ratio 3:2:1. The transition probabilities are computed as in Appendix B. Fig. 7(b) shows that SDCA can cope with the situation very well. (e) Eects of cell capacity: Although the SDCA algorithm is derived in the limit of large N , Fig. 8 shows that it is still very eective for N down to 40. (f) Eects of control period: To reduce signaling and computational load, it is desirable to increase the control period. Fig. 9 shows that SDCA has a steady performance when the control period is lengthened up to 80 s, roughly comparable to the dwell time of a call in a cell. However, in order to remain adaptive to sudden changes in network situations, the control period should not be lengthened inde nitely. (g) Eects of QoS requirement: Fig. 10 shows that SDCA is eective for a range of PQoS . (h) Eects of lower order local approximation: To demonstrate the relevance of the higher order terms used in computing the transition probabilities, we compare in Fig. 11 the results of including up to 1, 2 and 3 hops in the local approximation. Here we consider h = 0.05 s?1 and T = 20 s so that an average of one hando event per call takes place in a control period. It shows that multiple 11

hops are essential to a precise control. (i) Eects of reducing the distance of information exchange: The signi cance of multiple hops is relevant to the issue of reducing the signaling load in the network. We have based the control function on periodic exchange of status information among cells up to 3rd nearest neighbors. Restricting the distance of exchange to nearest neighboring cells will reduce the signaling load, but merely neglecting handos from cells beyond will inevitably sacri ce the control precision. Fig. 12 illustrates a mean- eld algorithm which maintains the control precision while restricting the distance of exchange. Here the transition probabilities are still computed up to 3 hops, with the status of the nearest neighboring cells being measured, whereas those of 2nd and 3rd nearest neighbors being assumed to take the average values of the nearest neighbors. Using the network with non-uniform trac in (d), the control performance essentially matches its measurement-based counterpart. The eectiveness of the mean- eld algorithm can be attributed to the fact that uctuations of the status of the twelve 2nd and 3rd nearest neighbors cancel out, improving the precision of estimating their average. Even for networks with non-uniform trac, the admission control maintains their occupancy and controlled arrival rate at about the same level. (j) Robustness against measurement errors: To study the robustness against measurement errors in the hando rate, we simulate a network whose trac is characterized by h = 0.01 s?1 , while the control is based on an inaccurate estimation hest . Fig. 13 shows that when hest diers from h by up to 20%, the dropping probability is still acceptable. For larger dierences, the error becomes unacceptable. Similarly, we study the robustness against measurement errors in the call arrival rate, by simulating a network whose trac is characterized by , while the control is based on an erroneous estimation of est = (1 + a) at each control period,  being a Gaussian variable with zero mean and unit variance. Fig. 14 shows that for a up to 20%, the performance is still acceptable.

5 Conclusion In this paper, we present a new distributed and dynamic call admission control scheme. The novelties of the proposed scheme which make it a stable and precise control are: (a) We have taken into account the eects of limited capacity and time dependence on the call dropping probability. (b) We have included the non-zero probability of multiple hops from distant cells for longer control periods. This improves the accuracy of the control mechanism. (c) Instead of implementing the control by adjusting the admission threshold, we have computed the acceptance ratio, which is able to spread the new calls uniformly over the control period. In contrast, adjusting the admission threshold tends to block late comers when it increases above the previous value, and early comers when it decreases below the previous value. The major advantage of SDCA is its insensitivity to the network load. The dropping probability is maintained at a stable level over a wide range of call rates. By comparison, the performance of DCA varies rapidly with the network load, and TR is designed for static control. Though there is a slight deviation from the prescribed PQoS in, say, Figs. 7 and 10, the stability of the algorithm implies that this can be oset easily. 12

The algorithm can operate over a wide range of networks: homogeneous and inhomogeneous call rates and hando rates, large and small cell capacities, long and short control periods, tight and loose QoS requirements, and erroneous measurements of call rates and hando rates. We have also shown that the signaling load can be further reduced by mean- eld estimations. Besides being adaptive to changes in the call rates, it also has the potential to be adaptive to changes in the hando rates using the local approximation for on-line updates. No matrix operations are required. The major complexity of the algorithm comes from solving (21) by the bisection method. However, since the precision for a stochastic control needs not be high (here we use a precision of 0.01 for ai ), this is not a problem for modern computers. While the dropping probability is an important parameter for network management, the probability of forced termination may be more relevant to the subscriber. If a mobile moves across K cells during its lifetime, the probability of forced termination is given by Pf = 1 ? (1 ? D)K where D is the average dropping probability. Though we have not presented results for the average probability of forced termination, it can be readily obtained by Pf = hD=(hD + ). Eorts are also in progress to extend the method for call admission control with multiple classes of trac, each having its own requirements bandwidth, QoS guarantee and hando rates. Given that our method is able to control the QoS precisely we expect that it has a high potential in the multiclass scenario.

Appendix A: The Transition Probabilities For Uniform Hando Rates We note that the matrix J can be written as

J = J0 + J1 ;

(22)

where (J0 )ik = (hk + )ik ; (J1 )ik =



?hik 0

k; i = nearest neighbors; otherwise:

(23)

For homogeneous hando rates, hk = h and hik = h=6. Considering J1 as the perturbation, we have [exp(?Jt)]ik =

1 (?t)r h   i J0r + J0r?1 J1 +


+ J1 J0r?1 +




X

r=0

r!

13

ik

:

(24)

The zeroth order term consists of those terms in (24) which contain no J1 . Hence the zeroth order contribution goes only to i = k, with

q0(t) =

1 (?t)r

X

r=0

r r! (J0 )ii = exp[?(h + )t]:

(25)

It corresponds to the case that no hando events take place between time 0 and t. The rst order terms consist of one and only one J1 . Since the elements of J1 are nonzero only for neighboring cells, the rst order contributions go only to neighboring cells i and k, with

q1 (t) =

1 (?t)r 

X

r=0

r!

(h + )r?1 h6 +


+ h6 (h + )r?1 = ht 6 exp[?(h + )t]: 

(26)

It corresponds to the event that a call is handed o from cell k to i. Higher order contributions can be evaluated similarly.

Appendix B: The Transition Probabilities for Inhomogeneous Hando Rates In the local approximation, the transition probabilities fik (t) from cell k to i consists of contributions from all possible paths starting from cell k and ending at cell i. Let P be a path of m hops following the sequence k ! l1 !


! lm?1 ! i. This path makes an mth order contribution to fik (t) given by

qP (t) =

1 (?t)r

X

r=0

r!

X

s0 +


sm =r?m

(hi + )sm hilm?1 (hlm?1 + )sm?1


hl1 k (hk + )s0 :

(27)

Since the path may visit a cell more than once, we denote the distinct cells along the path by the label j (including k and i), and the number of stops in each cell by mj . Hence after collecting terms corresponding to the same cell, (27) can be written as

qP (t) =

1 (?t)r

X

r=0

r!

2 X

Y

4

3 X

p1 +p2 +


=r?m j 2P s1 +


+smj =pj

(hj + )pj 5 hilm?1


hl1 k :

(28)

Consider the number of terms N (p; m) allowed in the summation s1 +


+ sm = p, where s1 ; s2 ;


are integers. It is equal to the number of ways of partitioning p indistinguishable objects into m 14

groups (empty groups allowed). Hence N (p; m) = (p + m ? 1)!=p!(m ? 1)!, leading to X

s1 +


+sm =p

@ m?1 (h + )p+m?1 : (h + )p = (m ?1 1)! @h m?1

(29)

Substituting into (27), we have

qP (t) =

1 (?t)r

X

r=0

2

3

@ mj ?1 (h + )pj +mj ?1 5 h 4 ilm?1


hl1 k : r! P pj =r?m j2P (mj ? 1)! @hjmj ?1 j j 2P X

1

Y

(30)

To simplify this expression, we use the following algebraic identities. Lemma

x1 : : : xr

1

: D(x1;


; xr )  : :

1

xr1?1 : Y r : = (xt ? xs): : t>s r xr?1













(31)

This identity can be veri ed directly by repeatedly applying row transformations to the determinant. Theorem X

n1 +


+nr =n

xn1 1


xnr r =

r

X

t=1

xtn+r?1 r (x ? x ) : s s6=t t

(32)

Q

The theorem can be proved by mathematical induction in r. First it can be easily veri ed for r = 2. Now assume that it holds for the value r, then for the value r + 1, we can write X

n1 +


+nr+1 =n

r+1 xn1 1


xnr+1

=

n

X

m=0

xnt ?m+r?1 xm ; Qr r+1 t=1 s6=t (xt ? xs ) r

X

!

(33)

where m = nr+1, and we have applied the theorem for the value r and the condition n1 +


+ nr = n ? m. The expression now reduces to a geometric series, and the result is X

n1 +


+nr+1=n

r+1 = xn1 1


xnr+1

r +1 xr?1 X xnr+1 xtn+r t Qr : + Qr +1 ( x ? x ) ( x ? x ) t s6=t (xt ? xs ) s t=1 r+1 t=1 s6=t t r

X

15

(34)

The rst term is just the desired terms up to t = r. In the second term, which will be denoted by T2 , we rst rearrange the factors appearing in the denominator for a given t as follows: r Y r ? t (x> ? x< ); (xt ? xs ) = (?) s6=t s6=t r

Y

(35)

where x> = xmax(s;t) and x< = xmin(s;t) . The factor (?)r?t appears since we haveQrearranged r ? t terms when t + 1  s  r. Multiplying both the denominator and numerator by ru>s6=t (xu ? xs), the second term in (34) becomes, on using (31),

T2 =

+1 xr?1 D(x ;


; x ; x ;


; x ) (?)r?t xnr+1 1 t?1 t+1 r : t xr+1 ? xt D(x1 ;


; xr ) t=1

r

X

(36)

We can rewrite T2 as

+1 T2 = xnr+1



1

: : :

1

x1 : : : xr









xr1?2 : : : r xr?2

xr1?1 =(xr+1 ? x1) : ?1 : : D (x1 ;


; xr ) : r ? 1 xr =(xr+1 ? xr )

(37)

This can be veri ed by expanding the determinant down the last column. Multiplying and dividing the tth row by xr+1 ? xt , and performing column operations successively, (34) reduces to n+r T2 = Qr (xxr+1 ? x ) : t t=1 r+1

(38)

Combining with the rst term in (34), the theorem is proved. We can now return to (30) which, on using (32), becomes

qP (t) =

1 (?t)r

X

r=0

r!

1 @ mj ?1 X mj ?1 j 2P (mj ? 1)! @hj j 2P Y

P

(hj + )r?m+[m? j (mj ?1)+1]?1 Q

t6=j (hj ? ht )

hilm?1


hl1 k ; (39)

where the number of distinct cells along the path is equal to m ? j (mj ? 1) + 1. The summation P

16

over r further reduces the expression to

qP (t) = (?)m

@ mj ?1 X exp (?(hj + )P t) Q m ? 1 j (m ?1) hilm?1


hl1 k : j 2P (mj ? 1)! @hj j 2P t6=j (hj ? ht )(hj + ) j j Y

1

(40)

Appendix C: Solution of the Diusion Equation Equation (12) with boundary condition (13) can be solved by Laplace transform, analogous to problems in heat conduction [3]. Let

F (x; z) =

Z

1

0

dte?zt P (x; t):

(41)

Using integration by parts and the initial condition (14), equation (12) can be transformed to !

@ ? D @ 2 F (x; z) = (x ? x ): z + v @x 0 @x2

(42)

Since (42) reduces to a homogeneous linear dierential equation for x < x0 and x > x0 , F (x; z ) can be obtained by piecewise construction, subject to the condition that F is continuous at x = x0 . The result is

F (x; z) =



Cek+(x?x0 ) C [(1 ? r)ek+(x?x0 ) + rek?(x?x0 )]

x < x0 ; x0 < x  1;

(43)

p

where k = (v  v2 + 4Dz )=2D. C is determined by the continuity condition that
@F=@x = ?1=D at x = x0,

C = Dr(k 1? k ) ; +

?

(44)

and r is determined by the boundary condition (13), 1 ? r exp[k (1 ? x )] = ? v ? Dk? exp[k (1 ? x )]: + 0 ? 0 r v ? Dk +

17

(45)

We are particularly interested in the solution at x = 1. Substituting (44) and (45) into (43), we nd "

p

#

2 exp ? v +24DDz ? v (1 ? x0 ) : F (1; z) = p 2 2 v + 4Dz ? v

(46)

The distribution P (1; t) can be obtained by inverse Laplace transform,

P (1; t) =

dz ezt F (1; z); 2 C i

Z

(47)

where C runs from s0 ? i1 to s0 + i1 to the right of all singularities of F (1; z ). Case 1, v > 0: F (1; z ) has a pole at z = 0 and a branch cut terminating at z = ?v2 =4D. Evaluating the contour integral along the branch cut explicitly,

P (1; t) = Dv + J (x0 ; t);

(48)

p

where, after a change of variable to  = ?4Dz ? v2 =2D,

J (x0 ; t) =

Z

0

1 d

8D exp ? t (4D2 2 + v2 ) + v (1 ? x ) 0 2 4D2 2 + v2 4D 2D

[2D cos (1 ? x0 ) ? v sin (1 ? x0 )] : Though this is dicult to integrate, we note that @J (x0 ; t)=@x0 is simpler, since

(49)

2i

(1?x ?vt) @J (x0 ; t) = exp ?p 40Dt 1 ? x0 : @x0 Dt 4Dt h

(50)

Integrating and using J (1; t) = 0, we arrive at (15). Case 2, v < 0: In this case, there is no pole at z = 0. For the integral along the branch cut, we integrate @J (x0 ; t)=@x0 and using J (?1; t) = 0, we again arrive at (15).

References [1] A. S. Acampora and M. Naghshineh, \An Architecture and Methodology for Mobile-Executed Hando in Cellular ATM Networks," IEEE Journal on Selected Areas in Communications, Vol. 12, No. 8, October 1994. 18

[2] B. Epstein and M. Schwartz, \Reservation Strategies for Multimedia Trac in a Wireless Environment," IEEE 45th Vehicular Technology Conference, Chicago, IL, July 1995. [3] A. L. Fetter and J. D. Walecka, Theoretical Mechanics of Particles and Continua, McGraw-Hill, New York, 1980. [4] D. Hong and S. S. Rappaport, \Priority Oriented Channel Access for Cellular Systems Serving Vehicular and Portable Radio Telephones," IEE Proceedings, Vol. 136, Part I, No. 5, October 1989. [5] S. M. Jiang, Danny, H. K. Tsang and B. Li, \Subscriber-Assisted Hando Support in Multimedia PCS," ACM Mobile Computing and Communication Review, Vol. 1, No. 3, September 1997. [6] N. G. van Kampen, Stochastic Processes in Physics and Chemistry, North Holland, Amsterdam, 1981. [7] D. A. Levine, I. F. Akyildiz and M. Naghshineh, \A Resource Estimation and Call Admission Algorithm for Wireless Multimedia Networks Using the Shadow Cluster Concept," IEEE/ACM Transactions on Networking, Vol. 5, No. 1, February 1997. [8] B. Li, C. Lin and S. Chanson, \Analysis of a Hybrid Cuto Priority Scheme for Multiple Classes of Trac in Multimedia Wireless Networks," ACM/Baltzer Journal of Wireless Networks, Vol. 4, No. 4, August 1998. [9] J. Misic, S. T. Chanson and F. S. Lai, \Admission Control for Wireless Multimedia Networks with Hard Call Level Quality of Service Bounds", Computer Networks and ISDN Systems, to appear in 1999. [10] M. Naghshineh and M. Schwartz, \Distributed Call Admission Control in Mobile/Wireless Networks," IEEE Journal on Selected Areas in Communications, Vol. 14, No. 4, May 1996. [11] E. C. Posner and R. Guerin, \Trac Policies in Cellular Radio That Minimize Blocking of Hando Calls," The 11th ITC, Kyoto, Japan, September 1985. [12] W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes in C: The Art of Scienti c Computing, Cambridge University Press, Cambridge, 1990. [13] R. Ramjee, R. Nagarajan and D. Towsley, \On Optimal Call Admission Control in Cellular Networks," IEEE Infocom'96, San Francisco, March 1996. [14] T. S. Rappaport, Wireless Communications: Principles and Practice, Prentice Hall, 1996. [15] M. Schwartz, \Network Management and Control Issues in Multimedia Wireless Networks," IEEE Personal Communications, Vol. No. June 1995. [16] S. Wu, K. Y. M. Wong and B. Li, "A New, Distributed Dynamic Call Admission Policy for Mobile Wireless Networks with QoS Guarantee", Ninth International IEEE Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'98), Boston, September, 1998.

19

(a)

8 18 7

+ 12

+...

3 11

1 6

16

+ 6

10

2

17

(b)

9

19

(c)

+ 2

+ 4

4 5

15

12

+5

13

+

+5

+...

14 Figure 1: (a) A 19-cell cellular network. (b) Topology of paths connecting k = i. (c) Topology of paths connecting k, i = nearest neighbors.

20

(a)

pn(0)

pn (t)

(b)

σ 0 pn (T)

n0

N

0 pN (t)

(c)

0

n0 N (d)

N

T

t

Figure 2: Schematic diagram of the channel occupancy distribution at (a) t = 0, (b) a subsequent time t > 0. (c) The nite capacity N causes a deviation from the Gaussian tail. (d) The evolution of the dropping probability pN (t).

0.07 0.06

Dropping Prob.

0.05 DCA 0.04 0.03 TR 0.02 SDCA 0.01 0.00

0

1 Call Rate per Second

Figure 3: The dropping probabilities for uniform trac.

21

2

1.0 DCA 0.8

Blocking Prob.

SDCA 0.6 TR 0.4

0.2

0.0

0

1 Call Rate per Second

2

Figure 4: The new call blocking probabilities for uniform trac.

TR

1.0

Channel Utilization

0.8

SDCA

0.6 DCA 0.4

0.2

0.0

0

1 2 Call Rate per Second

Figure 5: The channel utilizations for uniform trac.

22

3

0.05

0.04

Dropping Prob.

TR6(0.02) 0.03

0.02 SDCA(0.01) 0.01 SDCA(0.02) TR6(0.01) 0.00

0

1 Call Rate per Second

2

Figure 6: The dropping probabilities for SDCA and TR when the hando rate changes from 0.01 to 0.02 s?1 .

0.02

Dropping Prob.

(a)

0.01 (b)

0.00

0

1 Call Rate per Second

2

Figure 7: The dropping probabilities for (a) non-uniform trac, (b) inhomogeneous hando rates.

23

0.02

Dropping Prob.

N=40

N=70 0.01 N=100

0.00

0

1 Call Rate per Second

2

Figure 8: The dropping probabilities for cell sizes N = 100, 70, 40.

0.02

Dropping Prob.

T=10s

T=20s 0.01

T=80s

0.00

0

1 Call Rate per Second

2

Figure 9: The dropping probabilities for control periods T = 10, 20, 80 s.

24

Dropping Prob.

10

10

10

10

−1

−2

−3

−4

0.0

1.0 2.0 Call Rate per Second

3.0

Figure 10: The dropping probabilities for QoS requirements PQoS = 0.02%, 1%, 5%.

0.05 1−hop

Dropping Prob.

0.04

0.03

0.02 2−hop 0.01 3−hop 0.00

0

1 Call Rate per Second

2

Figure 11: The dropping probabilities for including up 1, 2, 3 hops in the local approximation.

25

Dropping Prob.

0.02

0.01

0.00

0

1 Call Rate per Second

2

Figure 12: The dropping probabilities for the mean- eld algorithm (symbols) and the measurementbased algorithm (line) in a network with non-uniform trac.

Dropping Prob.

0.02

0.01

−20% 0% +20%

0.00

0

1 Call Rate per Second

2

Figure 13: The dropping probabilities for erroneous measurements of the hando rate.

26

Dropping Prob.

0.02

0.01

30% 20% 0%

0.00

0

1 Call Rate per Second

2

Figure 14: The dropping probabilities for erroneous measurements of the call arrival rate.

27