Optimal Call Admission to a Mobile Cellular Network Mohammad Saquib & Roy Yates3 WINLAB ECE Dept., Rutgers University PO Box 909, Piscataway NJ 08855-0909
[email protected] [email protected]
Abstract To ful ll the future demand for and increase the reliability of mobile cellular service, we have explored the use of a call admission policy that constitutes a dynamic channel allocation scheme for a cellular system. The call admission policy optimizes a weighted blocking criteria for a queueing network model of a mobile cellular system. In this context, the call admission problem is formulated as a Markov decision process. The value iteration method has been applied to a uniformized chain of a cellular highway system. The e ectiveness of a call admission policy for maximizing the call completion rate and for reducing the hando dropping is studied. Channel reservation policies for hando s are also examined for di erent system parameters. In addition, several symmetry properties of the optimal policies which e ectively truncate the size of policy space, are mathematically established.
1
Introduction
The limited frequency bandwidth of the mobile cellular systems made apparent the inability of current wireless technology to meet future service requirements. A challenge for cellular engineers will be to meet customer demand by optimizing the existing resources. When a call moves from cell to cell, hando s are required. Due to the scarcity of channels, the new cell may not provide the incoming call with a channel. As a result, that hando will be dropped. From reliability of service point of view, this situation is not desirable and special actions need to be taken to prevent hando dropping. This kind of preferential treatment can be provided by a call admission policy. 3 This work was supported by NSF grant 92-06148 NCRI.
A call admission policy is the set of decisions that indicate when a new call will be served with a channel and when an existing call will be denied a hando from one cell to another. The simplest form of all policies is \admit all policy (AAP)"{ a call will be denied if and only if there is no available channel. On the other hand, we can assume that access requests or hando s can be regulated by an intelligent controller which has the ability to deny the call even when one (or more than one) channel is available. The purpose of rejecting the call is to optimize a system objective function such as average revenue or utilization of resources. Call admission policies for multiple class queues have been studied in [3,4,10,11]. In [7], analytical performance of call admission policies is numerically investigated for a seven cell system without hando s. In [9], the optimal call admission policy for a single cell of a cellular mobile network was studied. The policy iteration method of MDP [5] was used as a policy optimization technique. In that study, two distinct and independent classes of calls were considered in the system and the e ectiveness of call admission in the context of i) optimizing weighted blocking criteria and ii) maximizing the revenue of the system were evaluated.
1.1 A Model of a Cellular Network To perform a quantitative study of the optimal call admission policy for the cellular network, several assumptions are used. The cellular network is considered as single class system where all the calls in the system have the same stochastic properties. A highway cellular network or a city cellular network can be considered as two distinct single class systems. We assume that the call arrival process forms a stationary Poisson process with rate (calls/sec). This call holding time is drawn from an exponential distribution with mean 1= (sec/call). The average channel holding time of a call
in a cell before requesting a channel for hando is also assumed as exponentially distributed random variable with mean 1= (sec/call). The probability that the call in cell i requesting a hando goes to cell i + 1 is pi and pi = 1 0 pi is the probability that the call in cell i requesting a hando goes to cell i 0 1. The normalized o ered load of the calls is = n=c. The current state of the system will be represented by a vector x = (x1; . . . ; xn ), where xi is the number of calls in cell i and n is the number of cells in the system. The number of available channels is denoted by c. We require a re-use distance of 2. That is, a channel used in cell i can not be used in either cell i 0 1 or i + 1. As a channel allocation scheme this work considers Maximum Packing, MP, a dynamic channel allocation scheme described in [8]. The MP is analytically tractable and also in some sense optimal [1,2]. The MP can rearrange the channels in use in any way to accept a call, subject to the re-use constraints. We de ne clique sum, si (x), as the sum of active calls in cell i and cell i +1. That is, si (x) = xi + xi+1 . The set of feasible states, 3 = fx : sk (x) c; k = 1; . . . ; n 0 1g. When the system is at state x, a reject/accept decision must be made for each arrival and hando in each cell. Each decision variable can be either 1 or 0, where 1 stands for accept and 0 for reject. In state x, the decision variables for the cell i will be ai (x) for new the call, h+i01 (x) for the hando from cell i 0 1 and h0i+1 (x) for the hando from cell i + 1. Note that in a ring system with n cells, the cells are numbered 0; 1; . . . ; n 0 1. In this case, addition and subtraction of cell indices is modulo n. That is, when i = n 0 1, cell i + 1 = 0 is neighbor of cell i. If there is any violation of any of the clique sums, the system will not provide a new channel to the new call or hando , which means that call will be blocked or dropped immediately. Blocked calls and dropped calls are cleared. In addition, to ensure a valid channel assignment scheme, the clique sum requirements imply n can be even or odd for a linear system, but for a ring system, n must be even [13]. Let us de ne the complementary decision variables ai (x), h+i01 (x) and h0i+1 (x). At state x, the action space (the set of all possible actions for the new call and hando ) for cell i is ai (x) = fh+i01 (x); ai(x); h0i+1(x)g. The action space for the whole system at state x becomes, a(x) = fa1(x); . . . ; ai (x); . . . ; an (x)g. When the system is at state x 2 3 and the action a 2 a(x) is chosen, r(x; a) is interpreted as the reward rate, the expected time until a new state is entered is given by
P
P
(x; a) = [ ni=1 ai (x) + ni=1 xi ( + )]01 . If i (x) denotes the rate at which hando s from cell i at state x is dropped then i (x) = [h+i (x)pi + h0i (x)pi ] . Let I denote the set of cells (1; . . . ; n) and let ei denote a vector of zeroes, except for a one for the ith component. If Pxay , y 2 3, is the probability of the next state is y given that current state is x and the action a 2 a(x) is chosen, then for i 2 I 8 y = x + ei > ai (x) (x; a) > < f + i (x)gxi (x; a) y = x 0 ei Pxay = h+i (x)xi pi (x; a) y = x + ei+1 0 ei 0 > y = x + ei01 0 ei > : 0hi (x)xi pi (x; a) otherwise The value iteration algorithm of Markov decision process [12] has been used throughout as a technique for searching for optimal policies. Markov decision process, MDP has computational complexity that increases with the increase of the capacity of the system, the number of cells and the number of classes of customers. In [10], a technique to decompose the iterative equation of value iteration algorithm when the reward rate does not depend on the action was shown. This kind of decomposition of the iterative equation remarkably enhances the speed of the value iteration algorithm. To express an extended version of this decomposition technique, we consider tha above arbitrary linear cellular system. The value iteration algorithm for a continuous-time Markov chain is modeled using the uniformization technique in [12], for all x 2 3
3 2 X Vm (x) = max 4r(x; a) + P xay Vm01 (y )5 a2a(x) y23
(1)
where Vm (x) denotes the maximum total expected cost with m periods left to the time horizon when the current state is x. The uniformized transition probability is P xay which is derived from P Pxay and the uniformized rate 0 = [n + maxx23 ni=1 xi ( + )]. To solve the recursive equation 1, at each iterative step requires us to evaluate 23n possible candidate actions. In this work, we use a generalization of the decomposition of [10]. In particular, if the reward rate of each state x is an additive function of the action a(x), we can use the value iteration algorithm to nd the optimal policy by 3n evaluations over two actions in each iterative step. To demonstrate this decomposition, we can write the weighted reward rate for the current state x when the action a is chosen. Here we assign hando dropping a weight (always ! 1) where weight ! is interpreted as the relative penalty for dropping a hando
0.0
10
over blocking a new call. The weighted reward function becomes n X i=1
0xi
! p h+ (x)] i i
π
[0xi ! pi h0i (x) 0 ai (x) (2)
In addition, maximization of the call completion rate is equivalent to minimization of the weighted call incompletion rate with weight ! = 1:0.
Vm (x) =
X n
0 ( + xi! ) + Vm01(x) i=1
+(1=0)
n X i=1
[xi fVm01 (x 0 ei ) 0 Vm01 (x)g
+ai (x)fVm01(x + ei ) 0 Vm01 (x) + 0g +h0i (x)xi pi fVm01 (x 0 ei ) 0 Vm01 (x)g +h+i (x)xi pi fVm01 (x 0 ei ) 0 Vm01 (x)g +h0i (x)xi pi fVm01 (x + ei01 0 ei ) 0Vm01 (x) + !0g +h+i (x)xi pi fVm01 (x + ei+1 0 ei ) 0Vm01 (x) + !0g] (3) The decision variables ai (x) and h+i (x) can be expressed in terms of the value function Vm (x) as
ai (x) = 1[Vm01 (x + ei ) + 0 > Vm01 (x)]
(4)
h+i (x) = 1[Vm01 (x + ei+1 0 ei ) +! 0 > Vm01 (x 0 ei )] (5) where 1[1] is an indicator function and h0i (x) will be similar to h+i (x). Theorem 1 When ! > 1, if at state x a new call in cell i is accepted, then in state x + ei01 a hando from cell i 0 1 to cell i will be accepted. When ! = 1:0, a new call arriving at cell i in state x is admitted i a hando from cell (i 0 1) to cell i in state x + ei01 is allowed. Proof: For cell i at an arbitrary state s, we can express the hando decision variable as h+i01 (s) = 1[Vm01(s + ei 0 ei01 ) +! 0 > Vm01 (s 0 ei01 )] (6) If we rewrite the above policy h+i01 (s) for s = x + ei01 , then h+i01 (x + ei01 ) = 1[Vm01(x + ei ) +! 0 > Vm01 (x)] (7) Equation 4 and 7 prove the theorem.
Performance measures, B
r(x; a) =
-1.0
10
AAP
B OP B
-2.0
10
1.0
2.0 3.0 4.0 Normalized load, ρ (= nλ/cµ)
5.0
Figure 1: Single class linear system: Maximized call completion rate under variable normalized load with n = 3, c = 15.
2
Empirical Performance Measures
In this section, we shall observe the e ectiveness of the optimal policy in the context of a single class linear and a ring cellular system under variable system parameters. Let cc denotes the average rate (calls/sec) of call completion under the policy , where b and d are the average rate of new call blocking (calls/sec) and average rate of hando dropping (calls/sec) respectively. In this study, for a system with mobile calls, it is assumed that an arbitrary hando of a cell can go to either of the adjacent cells with equal probability. The probability of new call blocking under the policy will be B = b =n and the probability of hando dropping under the policy will be D = d =n. Probability of hando dropping is considered as the measure of the quality of service. However, we can not minimize only the hando dropping rate because the probability of hando dropping can be made zero by blocking every new call. Therefore, we weight hando dropping by a factor ! > 1; see equation 3. The weighted performance measures under the policy will be = B + !D . If the admit all policy, AAP, is considered as the stationary policy , then B and D will be B AAP and DAAP respectively. For the optimal policy B and D will be B OP and DOP respectively.
2.1 Maximizing Call Completion Rate If the linear system wants to support an incoming call at the end cells, then it needs to spend one channel which is shared by only one neighbor. So, supporting a call in the end cells is more ecient use of a channel than supporting a call in an arbitrary cell. As a result, under
-1.0
2.2 Optimizing Weighted Reward Rate In the previous section, we have proven that when the system wants to maximize the call completion rate, then it has to treat hando s and new calls equally. Under this restriction, the system can not follow a policy which is very di erent from the admit all policy when the calls are mobile. As a consequence, at higher mobility, we have seen that the probability of hando dropping is much higher than probability of call blocking, see Figure 2 at ! = 1:0. As expected, the system will decrease the probability of hando dropping with the increase of ! , see Figure 2. This will be achieved by saving more channels for the hando s, and as a consequence, the system will reject more new calls. If the system parameters are constants, then by selecting a proper value of the weight ! , the system providers can ensure any particular value of probability of hando dropping. Selection of ! implies selection of the sta-
-2.0
10
AAP
B OP B AAP D OP D
-3.0
10
1.0
2.0
3.0
4.0 5.0 6.0 7.0 Weighting factor, ω
8.0
9.0
10.0
Figure 2: Single class ring system: Optimal weighted call incompletion rate under variable weighting factor with n = 4, = 2:0, = 1:0, = 10:0, c = 10. 1.00
AAP
Ω * Π Ω r OP Ω
Weighted performance measures
the optimal policy the cells at the two ends never block a call when a channel is available. This situation also implies that the cells at the two ends have the lowest probability of new call blocking among all cells in the system. For the n cell linear system where n is large, we shall see that by the optimal policy, cells 2 and n 0 1 block calls more frequently. This happens because the optimal policy prefers to serve the calls in the end cells. The blocking characteristics of cells closer to the center are insigni cantly a ected by the preferential treatment which the optimal policy provides to the end cells. For small n, this preferential treatment given to the end cells has noticeable but not dramatic e ect on the overall system performance; see Figure 1. As the number of cells in the linear system increases, the ends e ect become insigni cant. In order to examine the e ectiveness of call admission separate from these end e ects, we examine a ring system where all cells face identical interference constraints. All of our following experiments will concentrate on a ring system of size four with the hope that the e ectiveness of call admission policy for a ring system of size four will express the e ectiveness of call admission policy in the context of an in nitely large linear system as well as a ring system of any size. In the previous section, we have noted that for achieving a valid channel assignment scheme, when channel re-use distance is 2, then the number of cells in the ring system should be even and four is the minimum number for the size of a ring system.
Performance measures
10
0.10
0.01 1.0
2.0
3.0
4.0 5.0 6.0 7.0 Weighting factor, ω
8.0
9.0
10.0
Figure 3: Single class ring system: Performance of optimal channel reservation policies for hando s under variable weighting factor when n = 4, = 2:5, = 1:0, = 10:0, c = 10. tionary policy associated with the weight. As a function of the hour to hour load, we can adjust ! to ensure a desired level of quality of service.
2.3 Optimal Channel Reservation Policies for Hando s For a xed channel allocation scheme, in any cell with capacity c^, a new call is accepted under a channel reservation policies, CRP, when the number of available channels for the cell is greater than r [6]. Here r denotes the number of reserved channels for hando s, 0 r c^. We extend this idea to DCA as follows: The decision variable for a new call at cell i at state x is ( c 0 maxfsi01 (x); si(x)g > r ai (x) = 01 otherwise Note that a call is admitted i there are more than r channels available in the cell i. If we consider the weighted performance measures as a function of r, then the optimal value of r, r3 will be de ned as
References [1] D. E. Everitt and N. W. Macfadyen. Analysis of Multicellular Mobile Radiotelephone System with Loss. Britt. Telecommun. Technol. J., 1(2):37{45, Oct. 1983.
-1.0
Performance measures
10
[2] D. E. Everitt and D. Man eld. Performance Analysis of Cellular Mobile Communication Systems with Dynamic Channel Assignment. IEE Select. Areas Commun,, 7:1172{1180, Oct. 1989.
-2.0
10
Π
*
B r OP B * Π D r OP D
-3.0
10
1.0
2.0
3.0
4.0 5.0 6.0 7.0 Weighting factor, ω
8.0
9.0
10.0
Figure 4: Single class ring system: Performance of optimal channel reservation policies for hando s under variable weighting factor when n = 4, = 2:5, = 1:0, = 10:0, c = 10.
[3] G. J. Foschini and B. Gopinath. Sharing memory optimally. IEEE Trans. Commun., 31(3):352{360, 1983. [4] I. S. Gopal and T. E. Stern. Optimal call blocking policies in an integrated services environment. In Conference on Information Sciences Systems, pages 383{388. Johns Hopkins University, 1983. [5] R. A. Howard. Dynamic Programming and Markov Processes. M.I.T. Press, 1960.
r3 = arg 0min
r rc = arg min fB r + !Dr g 0rc
(8)
where the stationary policy r is provided by the definition of the CRP for a particular value of r. When r = r3 , then r = r3 , the optimal CRP for hando s. Performance of the optimal CRP is observed in Figure 3 and Figure 4. Here it is also noted that the policy which minimizes the weighted call incompletion rate does not signi cantly outperform the optimal channel reservation policies for hando s. The optimal policy reserves sucient number of channels for hando s where it is necessary, while the optimal CRP always forcefully reserves r3 number of channels for hando s.
3
Conclusion
The policy which maximizes the call completion rate does not di er signi cantly from the admit all policy. Instead, this policy exhibited high probability of hando dropping at higher speed. Low probability of hando dropping is a measure of quality of service. Therefore, the weighted call incompletion rate in which the additional weight was given to the hando dropping was minimized. In this context, the optimal channel reservation policy for hando s was also examined. Increasing the value of weight in minimization of weighted call incompletion rate reduces hando dropping by blocking a higher number of new calls and the optimal channel reservation policies for hando s performs close to the optimal policy.
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a com-
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