Fairness in cellular mobile networks

University of Pennsylvania ScholarlyCommons Departmental Papers (ESE) Department of Electrical & Systems Engineering August 2002 Fairness in Cellular Mobile Networks Saswati Sarkar University of Pennsylvania, [email protected] Kumar N. Sivarajan Tejas Networks Follow this and additional works at: htp://repository.upenn.edu/ese_papers Recommended Citation Saswati Sarkar and Kumar N. Sivarajan, "Fairness in Cellular Mobile Networks", . August 2002. Copyright 2002 IEEE. Reprinted from IEEE Transactions on Information heory, Volume 48, Issue 8, August 2002, pages 2418-2426. Publisher URL: htp://ieeexplore.ieee.org/xpl/tocresult.jsp?isNumber=21938&page=1 his material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permited. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it. his paper is posted at ScholarlyCommons. htp://repository.upenn.edu/ese_papers/27 For more information, please contact [email protected]. Fairness in Cellular Mobile Networks Abstract Channel allocation algorithms for channelized cellular systems are discussed from a new perspective, viz., fairness of allocation. he concepts of relative and absolute fairness are introduced and discussed. It will be shown that under certain reasonable assumptions, there exists an absolute (max-min) fair carried traic intensity vector (a vector describing the traic carried in the cells of the system). We also show that this vector is unique. We describe some properties of the max-min fair carried traic intensity vector in an asymptotic limit where the traic and the number of channels are scaled together. For each traic patern, we determine a ixed channel allocation which atains this max-min fair carried traic intensity vector independent of the value of the ofered traic, in the same asymptotic limit. Finally, we discuss a tradeof between being max-min fair and trying to maximize revenue. We conclude this correspondence by discussing some possible extensions of our work. Keywords Asymptotic analysis, cellular networks, channel assignment algorithms, fairness, max-min fairness, revenue maximization Comments Copyright 2002 IEEE. Reprinted from IEEE Transactions on Information heory, Volume 48, Issue 8, August 2002, pages 2418-2426. Publisher URL: htp://ieeexplore.ieee.org/xpl/tocresult.jsp?isNumber=21938&page=1 his material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permited. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it. his journal article is available at ScholarlyCommons: htp://repository.upenn.edu/ese_papers/27 2418 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 8, AUGUST 2002 Fairness in Cellular Mobile Networks Saswati Sarkar, Member, IEEE, and Kumar N. Sivarajan, Member, IEEE Abstract—Channel allocation algorithms for channelized cellular systems are discussed from a new perspective, viz., fairness of allocation. The concepts of relative and absolute fairness are introduced and discussed. It will be shown that under certain reasonable assumptions, there exists an absolute (max-min) fair carried traffic intensity vector (a vector describing the traffic carried in the cells of the system). We also show that this vector is unique. We describe some properties of the max-min fair carried traffic intensity vector in an asymptotic limit where the traffic and the number of channels are scaled together. For each traffic pattern, we determine a fixed channel allocation which attains this max-min fair carried traffic intensity vector independent of the value of the offered traffic, in the same asymptotic limit. Finally, we discuss a tradeoff between being max-min fair and trying to maximize revenue. We conclude this correspondence by discussing some possible extensions of our work. Index Terms—Asymptotic analysis, cellular networks, channel assignment algorithms, fairness, max-min fairness, revenue maximization. I. INTRODUCTION AND SUMMARY In a cellular system, the coverage area is logically divided into cells. Each cell has a cell site or a base station. The communication from the mobile user is directed to a central switching office by the base station. The central switching office directs this communication to the destination. Depending on the mode of multiple access used by the mobile customers, cellular systems can be broadly classified into channelized and nonchannelized systems. In a channelized cellular system, the multiple access is time-division multiple access (TDMA) or frequency-division multiple access (FDMA), or a combination of both. The term channel refers to a time slot in TDMA, a frequency slot in FDMA, and a combination of both in TDMA/FDMA systems such as the Global System for Mobile Communications (GSM). Calls arrive and depart at random times in the cells of the system and a channel assignment algorithm must assign a channel to each call for its duration, while obeying certain channel reuse constraints. These reuse constraints can be modeled by a hypergraph [2], as explained in [7]. This correspondence will deal entirely with channelized systems. The cellular network operator whose task it is to choose an appropriate channel assignment algorithm is usually interested in maximizing his revenue and this amounts to maximization of the total traffic carried in the system, if we assume calls in all cells are charged at the same rate (dollars per unit time). In addition, in order to keep his customers satisfied, the operator has to provide them a minimum grade of service (maximum blocking probability). However, many channel assignment algorithms provide unequal grades of service in the various cells of the system, and are thus “unfair.” This correspondence attempts to study the concept of fairness in cellular networks. First, we shall motivate the study of fairness with an example. Manuscript received February 5, 1997; revised January 18, 2000. The work of S. Sarkar was supported in part by the National Science Foundation under Grant ANI01-06984. A portion of this work was performed at the Indian Institute of Science, Bangalore, supported under a grant from Nortel Networks. The material in this correspondence was presented in part at the 34th Annual Allerton Conference on Communications, Control, and Computing, Allerton, IL, September 1996, and published in the Proceedings of that conference. S. Sarkar is with the Department of Electrical Engineering, University of Pennsylvania, Philadelphia, PA 19104 USA (e-mail: [email protected]). K. N. Sivarajan is with Tejas Networks, Bangalore, 560 012, India (e-mail: [email protected]). Communicated by R. Cruz, Associate Editor for Communication Networks. Publisher Item Identifier 10.1109/TIT.2002.800495. Fig. 1. Three-cell system. Example: Consider the three-cell system shown in Fig. 1 and assume that a total of 80 channels are available. Adjacent cells are forbidden from using the same channel, but cells 1 and 3 can reuse the same channel. Consider two fixed channel algorithms, FCAA1 and FCAA2. FCAA1 allocates 40 channels to each cell and FCAA2 allocates 42 channels to cells 1 and 3 and 38 channels to cell 2. Let the offered traffic in each cell be 35 Erlangs. Let us assume that there are no handovers and no intercell calls. We also assume that neither algorithm allows any queuing of call requests. If there is a free channel in the cell in which a call is requested, the call request is honored; otherwise it is blocked. FCAA1 achieves a blocking probability of 5.4% in each cell and the total carried traffic is 99.3 Erlangs. For FCAA2, the blocking probability is 3.6% in cells 1 and 3 and 7.8% in cell 2. The total carried traffic is 99.8 Erlangs, slightly better than that of FCAA1. FCAA1 can be said to be “fairer” than FCAA2 since it treats the individual cells more “equally.” It is reasonable to assume that in this case a network operator will prefer FCAA1 to FCAA2 even though his total carried traffic is slightly reduced. Otherwise, he will risk the desertion of customers in cell 2 for a competitor operator. Thus, the problem of ensuring some level of fairness in channel allocation becomes an interesting one. We remark that fairness does not necessarily imply “equal blocking probability.” Various definitions of fairness are possible but the most appropriate one appears to be the notion of “max-min fairness.” A similar notion of fairness is widely used in the context of flow control [3]. Roughly speaking, max-min fairness minimizes the overall blocking probability without decreasing the blocking probability in any cell at the expense of other cells which are already worse off. More precise definitions follow in later sections. We make the following assumptions regarding the cellular system. The system consists of N cells and the underlying offered traffic model is independent from cell to cell; in particular, we ignore the effect of call handovers and intercell calls. However, it is likely that we can extend our results to the case in which this independence assumption is dropped and handovers and intercell calls can be included. Our optimism is derived from the fact that the results of [7], to which we shall refer extensively, have been extended to include handovers in [9]. The call requests form a Poisson process and the call duration is exponentially distributed. The N cells share a common set of n channels. If Ai denotes the offered traffic in cell i (the expected number of calls that would be in progress in cell i if all call requests could be honored), then Ai =n is the offered traffic intensity in cell i. The offered traffic intensity in the system, r , is the sum of the offered traffic intensities N Ai repin the cells, thus r = Ai =n. The ratio pi = Ai = i=1 resents the fraction of the total offered traffic in cell i and the vector p ~ = (p1 ; p2 ; . . . ; pN ) is the traffic pattern. The carried traffic intensity in cell i, xi , is the carried traffic (expected number of calls in progress) in cell i per available channel in the system. We call the vector x ~ = (x1 ; x2 . . . ; xN ) the carried traffic intensity vector. When a call request arrives in a cell, the channel assignment algorithm either assigns it a channel for its entire duration, or blocks it. (A blocked call 0018-9448/02$17.00 © 2002 IEEE IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 8, AUGUST 2002 2419 = Fig. 2. Scale diagram. In all three cases illustrated B is fairer than A. In (a) and (b) N 2. In (c) N = 5. For both A; B of (a) and (b) and A of (c) components are represented in order, with component 1 the lowermost. For B of (c) the order is 1, 2, 3, 5, 4, component 1 being the lowermost on the vertical line. Note that corresponding components of B are joined with those of A. The numbers are the values of the corresponding call acceptances. disappears from the system.) The decision regarding blocking or honoring a call request is taken by the channel assignment algorithm based entirely on the current value of a suitably defined state vector. Further, we assume that the state vector assumes a finite, or countably infinite, set of values. In a nutshell, we assume that channel assignment algorithms can be modeled by continuous-time Markov chains (CTMCs). Further, we assume that there exists a probability distribution on the states of the CTMC, which satisfies the balance equations and all state probabilities are positive.1 An example of a state vector is a list that specifies the calls in progress in each cell of the system and the channels assigned to them. The assignment of channels to calls at any stage must satisfy certain reuse constraints which essentially specify the sets of cells that may simultaneously use the same channel. Each possible channel assignment algorithm corresponds to one and only one carried traffic intensity vector under fixed r , p~, and n. We denote the set of achievable carried traffic intensity vectors under fixed r , p~, and n, by F (r; p; ~ n) or simply F . The blocking probability in cell i is bi = 1 0 xi =pi r . We denote ci as the call acceptance in cell i, where ci = xi =pi . c ~ = (c1 ; c2 ; . . . ; cN ) is denoted as the call acceptance vector. ci =r is the call acceptance probability in cell i. We consider only systems in which pi > 0 for all i.2 The carried traffic, blocking probability, and call acceptance probability depend on the channel assignment algorithm used and we shall denote this by an appropriate suA perscript. For example, x ~ is the carried traffic intensity vector under channel assignment algorithm A. We will proceed as follows. In Section II, we formally define the concept of relative fairness and present a simple necessary and sufficient condition for relative fairness. In Section III, we introduce the notion of absolute fairness and prove its existence and uniqueness. In Section IV, we deal with fairness in the asymptotic limit, i.e., when the number of channels is arbitrarily large. We also discuss the computation of the ab- solutely fair carried traffic intensity vector in the asymptotic limit and give the corresponding channel allocation strategy to attain it. In Section V, we discuss the loss of revenue brought about by fairness and discuss a tradeoff between being fair and maximizing the revenue. We conclude this correspondence by discussing how our work can be extended. 1This is equivalent to assuming the CTMCs to be positive, recurrent, and regular. 2This means that we eliminate from the system cells with no offered traffic. 3Scale diagram is a schematic representation of vectors. The components (call acceptances for the cellular examples) are represented as points on a vertical line. Corresponding components of vectors are joined. II. RELATIVE FAIRNESS Informally, a channel assignment algorithm A is fairer than another channel assignment algorithm B , for given r , p~, and n, if for every cell i whose blocking probability is decreased by B compared to A, there is some other cell j whose blocking probability was already no less than that of i under A and has been increased further by B . A more formal definition of relative fairness is as follows. A channel assignment algorithm A is fairer than another channel assignment algorithm B , under the same r , p~ and n if A 6 x~B and • x ~ = • if there exists an i such that biA > biB , then there exists a j such that bjA  biA and bjB > bjA . Equivalently, if there exists an i such that ciA < ciB , then there exists a j such that cjA  ciA and B A cj < cj . Since a channel assignment algorithm corresponds to a unique carried traffic intensity vector for fixed r , p~, and n, we will usually speak of the fairness of carried traffic intensity vectors rather than the fairness of channel assignment algorithms. Note that given any one of the vectors, x b = (b1 ; . . . ; bN ), and c ~ = (c1 ; . . . ; cN ), we can determine ~, ~ the other two uniquely (since r and p~ are fixed). Thus, we may also equivalently consider the fairness of the vectors ~b or c~ instead of x ~. Fig. 23 shows pairs of carried traffic intensity vectors. In each case, one is fairer than the other. 2420 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 8, AUGUST 2002 Theorem 1: A channel assignment algorithm A is fairer than another channel assignment algorithm B if and only if there exists an A = mini2S ciA and Amin 2 S  U = f1; 2; . . . N g such that cA B A B A > cA , where S = fi: ci 6= ci g. cA Remarks: In other words, this theorem states that the necessary and sufficient condition for fairness of a channel assignment algorithm A over another B is that the least call acceptance probability (over all cells, ignoring those with equal call acceptance probabilities under both A and B ) under A is strictly greater than the corresponding call acceptance probability in B . This theorem will be useful in discussing absolute fairness. Note that relative fairness is different from lexicographic ordering.4 If a vector is fairer than another, it is lexicographically greater as well. However, a vector may be lexicographically greater than another, but neither of the two may be fairer than the other. Thus, the necessary and sufficient condition in this theorem is sufficient but not necessary for lexicographic comparison. Proof of Theorem 1: The proof will be given in two steps. First, the sufficiency will be proved and then the necessity A Let cA B > cA Then for every B such that cA j A where cA ; 2 A , for which U A jA Amin 2 < c (1) S: jB , there exists Amin c and < c A = min ct ; t2S 2 = min U c If possible, let A A  c B A B < cA A cA 2 fo all Amin satisfying (1). Since A is fairer than B , there exists j A A j  cA A A S; c 6= B A c 2 U , for which (3) A B j > cj : (4) c For j 2 U nS , (cjA = cjB ) and, hence, inequality (4) is never satisfied for any j 2 U n S . For j of Amin in (1) 2 S , from inequality (3), and the definition A A j = cA c i.e., A t2S t = min c satisfies (1). Hence, from (2), (4) cannot be satisfied for any . Therefore, there is no j 2 U for which both inequalities (3) and (4) are satisfied and that is a contradiction. } j 2 j S We now prove another interesting result which we shall use in obtaining some results on absolute fairness. Lemma 1: If the channel assignment algorithm A is fairer than the channel assignment algorithm B , then B cannot be fairer than A. (In other words, fairness is an antisymmetric relation.) 4Two B A 2 such that AA B since  min i2S i S A > cB c A i2S i = min c c Amin 2 (5) S: Since B is fairer than A, by Theorem 1 Bmin B = cB > cA : c B B i2S i A A , since Bmin 2 S and by inmini2S ci = cA 2 S such that min A  Again cB equality (5). Combining the above inequalities we get A A c B A  > c . Hence, (2) c and Amin B B c A B > c  A A c : This is a contradiction. Hence the existence of two channel assignment algorithms mutually fairer than each other is not possible. c for all Amin satisfying (1). Since Amin 9 9 c jA 6= cjB , j 2 S .) Since Amin 2 S , S is nonempty and thus A ~B . Thus, A is fairer than B . x ~ 6= x A ~B and hence S is nonempty. Let A be fairer than B . Thus, x ~ 6= x (Since Proof of Lemma 1: Assume there exist channel assignment algorithms A and B , each fairer than the other. Since A is fairer than B , by Theorem 1 c A  cA : j t2S t A A c Fig. 3. Scale diagram. This figure compares the fairness of two vectors A and B with two components each. Neither A nor B is fairer than the other. For A, components are represented in order, with component 1 the lowermost. For B , the order is 2,1, component 2 being the lowermost on the vertical line. Note that the corresponding components of B are joined with those of A. The numbers are the values of the corresponding call acceptances. vectors can be lexicographically compared as follows. If the minimum components are unequal, then the vector with a larger minimum component is lexicographically greater. If the minimum components are equal, then the second minimum component must be considered, and so on. III. ABSOLUTE FAIRNESS A channel assignment algorithm is absolutely fair or max-min fair if it is fairer than any other channel assignment algorithm which achieves a different carried traffic intensity vector under the same r , p~, and n. We shall refer to the carried traffic intensity vector corresponding to the max-min fair channel assignment algorithm as the MMF carried traffic intensity vector, or simply the MMF. Informally, a carried traffic intensity vector is the MMF if the blocking (probability) in any cell under it cannot be decreased without increasing the blocking in any cell already experiencing greater or equal blocking. It is not obvious that every set of carried traffic intensity vectors F has an MMF. Consider the following examples. Example III.1: Let F consist of only the carried traffic intensity vectors corresponding to the vectors c~A and c~B shown in Fig. 3. Neither A nor B is fairer than the other. Hence, no MMF exists in F in either case. Example III.2: Let F = RN . (RN is the set of N -dimensional V vectors with real components.) For any vector x ~ 2 F , there exists C another vector, x ~ 2 F , each of whose components is greater than the V V C corresponding one in x ~ . Thus, x ~ is not fairer than x ~ and hence is not the MMF. The same observation applies for F = f~ = ( x x1 ; x2 ; ...; x N ): 0  xi < 1g: In view of the above examples, it becomes necessary to prove the existence of an MMF. In this section, we will first show that any set of vectors with certain properties has an MMF. The definition of fairness in any set of N -dimensional vectors remains the same as that for carried IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 8, AUGUST 2002 traffic intensity vectors. cVi = xVi =pi , where xVi , is the ith component of the vector V~ . p~ = (p1 ; p2 ; . . . ; pN ) is the same for the entire set. pi > 0, for all i 2 U . The necessary and sufficient condition given in Theorem 1 for relative fairness of channel assignment algorithms holds for that of vectors in any N -dimensional vector set as well. A vector is the MMF if it is fairer than all other vectors in the set. Next, we discuss whether the set of achievable carried traffic intensity vectors of a cellular system has these properties. We end this section with the result that the MMF is unique. Theorem 2: If P is a nonempty, coordinate convex,5 convex, closed, bounded subset of RN , P has an MMF. Proof of Theorem 2: For brevity, we only give a construction for a vector that can be proved to be max-min fair using Theorem 1. For details, refer to [11]. Unless otherwise stated, U = f1; 2; . . . ; N g. Construction: Start from the all-zero vector ci = 0, 8 i 2 U . Always find the largest subset Il  U such that ci ’s, i 2 Il , can be increased equally without decreasing any ci . Increase the ci ’s of this subset equally without decreasing any ci , till it is no longer possible to do so. The procedure terminates when the largest subset which can be increased equally without decreasing any ci is empty, i.e., no ci can be increased without decreasing some other ci . Informally, the procedure goes as follows: Start from the vector (0; 0 . . . ; 0). Suppose the largest subset which can be increased is f1; 2; . . . ; kg, where 1  k  N . Increase c1 ; . . . ; ck equally till a vector (p1 ; . . . ; pk ; 0; . . . ; 0) is reached and  0 is such that (p1 ( + ); . . . ; pk ( + ); 0; . . . ; 0) 62 P for any > 0. Again increase the largest possible subset equally without decreasing any component. The largest possible subset will be a proper subset of f1; . . . ; kg. Let it be f1; . . . ; mg, m < k . The increase of this subset continues till (p1 + ); . . . ; pm ( + ); pm+1 ; . . . ; pk ; 0; . . . ; 0) 62 P for any  > 0. The procedure terminates when no element can be increased without decreasing some other(s). Note that this is a generalization of the construction of max-min fair rate allocation presented in [3]. The construction in [3] applies to polytope feasible sets defined by linear constraints of the type, AX  B . The construction we present here applies to more general feasible sets. We discuss the validity of the assumptions that the set of carried traffic intensity vectors is nonempty, closed, bounded, convex and coordinate convex. Nonempty: The set of carried traffic intensity vectors is clearly nonempty. Closed: In an asymptotic limit where the number of channels and the offered traffic are made arbitrarily large while keeping the ratio finite, and which we consider in the next section, for the model we have assumed (underlying model of offered traffic independent from cell to cell, etc.), the set of carried traffic intensity vectors is given by the following inequalities (refer to [7]): xi  M j=1 subset of R (R negative components), 5A x~ = ( x ;x to F for all Xj aij ; i = 1; 2; . . . N (6) is the set of all N -dimensional vectors with real nonF , is said to be coordinate convex if for each vector ; . . . ; x ) in F , x~ = (x ; x ; . . . ; x ; . . . ; x ), belongs satisfying 0 < 1 and for all i 2 f1; 2; . . . ; N g.  M j=1 Xj  0 (7) Xj = 1 (8) xi  pi r; xi  0; i = 1; 2; . . . N i = 1; 2; . . . N (9) (10) where aij = 0 or 1 depending on the system configuration and channel reuse constraint. The aij can be found by modeling the system by a hypergraph [7]. This set is closed. For the finite channel case, we could not prove that the set of carried traffic intensity vectors F is closed. This remains an assumption. However, even if the assumption turns out to be invalid our result is not seriously weakened. It can be shown that the closure of any convex, coordinate convex, and bounded set is also convex, coordinate convex, and bounded and obviously closed. Also, clearly closure of a nonempty set is nonempty. We are going to show that the set of carried traffic intensity vectors is convex, coordinate convex, and bounded. Thus, the closure of the set of carried traffic intensity vectors is nonempty, convex, coordinate convex, bounded, and M (by Theorem 2). a closed subset of RN and hence has an MMF x ~ M Even if x ~ does not belong to F (closedness is not necessary for the M existence of MMF, x ~ may belong to F ), we have carried traffic inM tensity vectors in F arbitrarily close to x ~ . For practical purposes, it suffices to assume that the set of carried traffic intensity vectors is closed. Bounded: The set of carried traffic intensity vectors F is bounded. This is because the carried traffic intensity can neither exceed the offered traffic intensity in any cell nor can it exceed 1 (since the total number of channels available in the system is n, the traffic carried in any cell cannot exceed n) and it is also nonnegative. Thus, 0  xi  min(pi r; 1) for each i and hence 0  ci  min(r; 1=pi ) for each i, where 0 < pi < 1. Theorem 3: The set of achievable carried traffic intensity vectors for a cellular system F is convex. ; . . . ; pm ; pm+1 ; . . . ; pk ; 0; . . . ; 0) is reached and (p1 ( 2421 For brevity, we give an outline of the proof of this important result. Refer to [11] for details. Our assumptions about channel assignment algorithms, the call arrival process, and call duration distribution allow the operation of the channel assignment algorithms to be modeled by a CTMC. Furthermore, we assume the CTMCs to be positive recurrent A and irreducible. Let there be two carried traffic intensity vectors x ~ and B x~ realized by channel assignment algorithms A and B , respectively. Let SA = fA1 ; A2 ; . . . ; g (resp., SB = fB1 ; B2 ; . . . ; g) denote the state space of A (resp., B ). Denote one null state (state in which the number of calls in progress in the system is 0) of A by TA and one of B by TB . A third channel assignment algorithm C , whose state space is the union of SA and SB , operates as follows. When in TA it switches over to TB at rate AB and when in TB it switches over to TA at rate BA . Otherwise, C behaves in the same manner as A or B depending on whether it is in a state that belongs to SA or SB , respectively (all other transition rates remain the same). It can be shown that the CTMC C A ~ + corresponding to C is positive recurrent and irreducible. x ~ = x B A B 0 1 (10 )~ x , where = (1+ T =T ) , = AB = BA , and fA g and f B g are the steady-state probability distributions of algorithms A and B , respectively. By adjusting we can get any 2 (0; 1). Theorem 4: The set of achievable carried traffic intensity vectors for a cellular system F is coordinate convex. We need to prove that if there exists a carried traffic intensity vector x~A 2 F , then any vector of the form x~B A ; xA ; . . . ; i xi ; . . . ; xA ) 2 F 2 N = (x1 2422 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 8, AUGUST 2002 where 0  i < 1, for any i 2 U . A slight variation of the technique used to prove Theorem 3 in conjunction with Theorem 3 can be used to prove this theorem. Refer to [11] for details. Thus, the set of achievable carried traffic intensity vectors for a cellular system is closed, bounded, convex, and coordinate convex. It is also a subset of RN . Hence, it has an MMF by Theorem 2. Theorem 5: MMF is unique. Proof of Theorem 5: Let the max-min fair carried traffic intensity vector (MMF) not be unique. Hence, there exist at least two different M M MMFs, x ~ and x ~ . Thus, the corresponding channel assignment algorithms, M1 , M2 are both fairer than each other (~ xM 6= x~M ). This violates Lemma 1. Hence this is not possible. } IV. PROPERTIES AND COMPUTATION OF MMF IN THE ASYMPTOTIC LIMIT In this section, we shall consider an asymptotic limit where Ai and n ! 1 but !1 !1lim !1 A =n = p r A ;n i i is finite. Thus, both the offered traffic and the number of channels are made arbitrarily large while keeping the ratio—the offered traffic intensity—finite. The properties of channel assignment algorithms in this asymptotic limit were studied in [7]. The feasible set of carried traffic vectors can be described by some linear inequalities in this case. Incidentally, [5] studied a different notion for fairness, proportional fairness, for resource allocation in wireline case, and presented computational strategies for obtaining the same. Also, [5] showed that max-min fairness is a limiting case of a generalization of proportional fairness, and as such a max-min fair allocation can be approximated arbitrary closely using the techniques for computing proportionally fair allocation. The similarity between the wireline case considered in [5] and the asymptotic case for cellular networks considered in this section is that the feasible sets can be described by linear inequalities in both cases (the nature of the inequalities differ in the two cases though). The computational approaches differ in the two cases. The important advantage of our approach is that we exploit specific properties of max-min fair allocation and the feasible set in the cellular mobile case to develop parametrized closed-form expressions for MMFs and the corresponding channel allocations. The parameters in the closed-form expressions can be obtained by solving linear programs. The approach in [5] is to use a nonlinear optimization based iterative update technique for obtaining the proportionally fair allocation, and the iterative procedure is not guaranteed to converge in finite number of iterations. We present our results in what follows. M Let x (r ) denote the MMF carried traffic intensity vector at load r ~ in this asymptotic limit. In this limit, the set of achievable carried traffic intensity vectors F (r) at r is described by inequalities (6)–(10) for our model [7]. The proofs in this section use results from [7]. 1 i Theorem 6: There exist finite nonnegative f 2 U and r  0 c M i (r ) = min(r; i g such that for each V ~ may be the all-zero construction remains identical for some time. x vector) and then let the constructions differ (if the constructions never M M M M differ x (r1 ) = ci (r2 ) for all i 2 U (r1 ) = x ~ (r2 ), ci ~ and there is nothing to prove). Let the largest subset of components V of x ~ whose ci s can be increased equally without decreasing others be I1 and I2 at r1 and r2 , respectively. Since the constructions differ, henceforth I1 6= I2 . Let, if possible, I1 be nonempty. At any r , if I is V the largest subset of components of any vector x ~ 2 F (r ), whose ci ’s can be increased equally without decreasing any other component, V ~ with j in U n I can be increased without decreasing then no cjV of x any other component, else from convexity and coordinate convexity of F (r), all ci ’s of I [ fj g  I can be increased equally without V decreasing any other component. Thus, no component of x ~ in U n I1 (U n I2 ) can be increased at r1 (r2 ). Observe F (r1 )  F (r2 ) V for r1 < r2 . Thus, the ci ’s of x ~ in I1 can also be increased equally V ~ without decreasing others at r2 . Hence, I1  I2 . Since ci ’s of x in I2 n I1 can be increased at r2 , but not at r1 , without decreasing others, from inequalities (6)–(10), ciV = r1 , for i 2 I2 n I1 , but from construction principle at r2 , ciV = cjV for all i; j 2 I2 (till this point ci ’s of sets I2 have been increased equally and that is how x~V has been reached). Thus, ciV = r1 for all i 2 I2  I1 and no component V of x ~ in I1 can be increased at r1 (inequality (9)). Thus, I1 = , i.e., V M V construction at r1 has terminated and x ~ = x ~ (r1 ). Since x ~ is an intermediate vector in the construction procedure for MMF at r2 , from construction principle 1 1 c (r2 ) c M i V i c = (r1 ) M i 2 U and this part of the theorem is proved. for each i 2 U at r = 0. Thus, for each i 2 U , there exists some  0 (possibly 0) such that for all r  , c (r) = r. Clearly, for each i 2 U , there exists some finite r  0, such that for all i (r ) = 0 c M i i c M i (r ) c M i M i i < r (e.g., for r > M ). Let = i (r ) = r for r < of r , and c M i ( i) M i ( c  i) inf (r )<r r: c r < 1: . Since ciM (r) is a nondecreasing function i (inequality (9)) i 2 [ 0 ; ]; i for all  > 0: i M Hence, cM ( i ) = i . Consider any r1 > i , for which ci (r1 ) < i r1 (clearly, there exists at least one such finite r1 since i is finite). ConM V V sider any r2 > r1 . Let x (r2 )), ~ be a vector with cj = min(r1 ; cj M V for each j 2 U . Clearly, x ~ 2 F (r1 ). Since cj (r ) is a nondecreasing function of r , and cjM (r)  r (from inequality (9)) 1 c M j Since x ~ V (r1 )  min(r ; c 1 is not fairer than x ~ M j for each j 2 U and c from the definition of (r1 ) c M i (r1 ) = c ; c < r1 . Thus, c c M i M i 0 (r1 ) = M i c M i (r2 ) and i c  > 2 U: (r2 )) M j (r2 ); for all r1 ; Again, from the nondecreasing property of upper-bounded by r for each r > i. for each j V j (r1 ) (r1 ) = min(r1 ; M i : (r2 )) = M j M c i) Proof of Theorem 6: First we shall prove that ciM (r) is a nondecreasing function of r . Consider 0  r1 < r2 . We show that M cM (r1 )  ci (r2 ) for all i 2 U . Carry out the construction given i in proof of Theorem 2 for both, in the asymptotic limit. The outputs M M will be the respective unique MMF’s, x ~ (r1 ) and x ~ (r2 ). Let the construction procedures for both remain identical upto a certain V point, when the carried traffic intensity vector reached is x ~ (since construction for both start from the same vector, i.e., the null vector, the 1 1 (r ) and for any 2[ ; r > i i i c M i r2 > : i (r ), and since it is + ] . Thus, c M i (r ) = i , for all } The construction procedure given in proof for Theorem 2 can be M implemented using the inequalities (6)–(10) to yield x ~ (r ), at any IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 8, AUGUST 2002  0, as follows. Start from the null vector. Find the largest sibly 0), such that (p1 ; p2 ; . . . ; pN ) 2 F1 (r) r s: s  = max M j M j =1 Xa j X =1 i :  0; j = 1; 2; . . . ; M ; s  r X j = 1; (pos- p ; i = 1; 2; . . . ; N ; ij j 2423 V 2 F1 (r)g, where x~V is any carried traffic Next find I = fi: x ~ intensity vector of the form > ;  ; c V j j=i otherwise V (inequalities (6)–(10) may be used to test whether x ~ 2 F1 (r )). I is thus the largest subset whose call acceptances can be increased equally without decreasing any other component of (p1 ; p2 ; . . . ; pN ). Without loss of generality, let I = f1; 2; . . . ; kg, k < N . All call acceptances of I cannot be increased equally without decreasing any other component when a carried traffic intensity vector (p1 ; p2 ; . . . ; pk ; pk+1 ; . . . ; pN +1 ) is obtained such that (p1 ( +); for any p2 ( > +); . . . ; p +); k( p +1 ; . . . ; p +1 N k ) 62 F1 (r) Fig. 4. 0 s: s  M = max j M j X j =1 M j Xa j =1 = 1; X j Xa =1 ij j ij p ; i 2 I; i  p ; i 2 U nI ; i :  0; j = 1; 2; . . . ; M ; s  r V Again, I1 = fi: x ~ 2 F1 (r)g is the largest subset of components whose call acceptances can be increased without decreasing any other component of (p1 ; p2 ; . . . ; pk ; pk+1 ; . . . ; pN +1 ), where x~V is any carried traffic intensity vector of the of the form: c V j > ;  ;  ; j=i j 2 I n fig otherwise . The procedure terminates when no call acceptance can be increased M without decreasing others. The output will be x (r ). ~ M 0 If x ~ (r ) is found at some sufficiently large r , say r for which M M 0 0 0 ci (r ) < r , for all i 2 U , then ci (r ) = i , for all i 2 U , from 0 M Theorem 6. Thus, the i ’s can be determined from x (r ). Using ~ M these i ’s, x ~ (r ) can be determined for all r  0 from Theorem 6. M Thus, x ~ (r ) need not be found separately at all values of r . (Using inequalities (6), (8), and (9), for any r > 1= mini2U pi , ciV < r , for M V all i 2 U , and any x ~ 2 F1 (r ). Hence, ci (r ) < r for each i 2 U , if r > 1= mini2U pi .] Theorem 7: The carried traffic intensity vector corresponding to the fixed channel assignment algorithm which allocates bnpi i c channels to cell i, for all i 2 U , is the MMF for all r  0 in the asymptotic limit. Remark: This fixed channel allocation scheme is similar to the one used to attain the highest carried traffic intensity in [7]. First, we show that it is possible to allocate bnpi i c channels to each cell i at all r . For some rl  maxi2U i , ciM (rl ) = i , by Theorem 6 (rl is finite). Thus, xi = pi i satisfies inequality (6) at all r where fXj g of inequality (6) satisfies inequalities (7) and (8) at all r . From [7] bn M Xj aij c channels can be allocated to each cell i j =1 where fXj g satisfies inequalities (7) and (8). Thus, from inequality (6) bnpi i c channels can be allocated to each cell i at all r. Again from [7], in the asymptotic limit any allocation which gives bnfi c channels to a cell i attains a carried traffic intensity Nineteen-cell system. of min(fi ; ri ) in cell i, if offered traffic intensity in cell i is ri for the model we have assumed. Thus, the algorithm allocating bnpi i c channels to each cell i attains a carried traffic intensity of pi min( i ; r) in each call i in the asymptotic limit, for all r , i.e., } ci (r) = min( i ; r) = ciM (r) at all r (Theorem 6). Example IV.1: Consider the system of 19 cells (see Fig. 4) with the following reuse constraint. A set of cells can use the same channel simultaneously if the interference in each cell of the set is  3=8. The interference in any cell u is given by v2S; v 6=u d(u; v )04 , where S is the set of cells using the same channel simultaneously and d(u; v ) is the distance between the centers of cells u and v . We assume the cell p radius to be 1= 3 or equivalently, the distance between the centers of adjacent cells to be unity. (This system and reuse constraints are the same as those in [7, Example 1.2].) 1) We shall consider uniform traffic, i.e., pi = 1=19 for each cell i 2 U . The carried traffic intensity in the system under an MMF is given by T M (r ) = N =1 x M i (r ) = min(r; 247=49) i (r ) = min(r=19; 13=49), 8 i 2 U . (see Fig. 6 (a)). xM i Thus, ciM (r) = xM (r )=pi = min(r; 247=49), 8 i 2 U . i In Fig. 5(a), these call acceptances have been plotted. A fixed channel allocation algorithm allocating b13n=47c channels to each cell achieves the MMF carried traffic intensity vector at all r in the asymptotic limit. 2) Let us consider nonuniform traffic 1=24; p i = 1=16; 1=8; i 2 f1; 2; 3; 4; 7; 8; 12 13; 16; 17; 18; 19g i 2 f5; 6; 9; 11; 14; 15g i = 10. and This pattern of nonuniform traffic resembles that in cities in which traffic is maximum in the central portion and decreases as we move toward the outskirts. The max-min fair carried traffic intensity in the system TM (r) = min(r; 2 + r2 ; 13 3 ) (refer to Fig. 6 (b)]. 2424 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 8, AUGUST 2002 xM i (r ) = i= pi min(r; i ), 8 i 2 U . 14=3; 4; i 2 f1; 2; 3; 4; 7; 8; 12; 13; 16; 17; 18; 19g otherwise. In Fig. 5(b), call acceptances have been plotted for these cells. The following fixed channel allocation algorithm gives the MMF at all r in the asymptotic limit (ni is the number of channels allocated to the ith cell): b7n=36c; ni = bn=4c; bn=2c; (a) i 2 f1; 2; 3; 4; 7; 8; 12; 13; 16; 17; 18; 19g i 2 f5; 6; 9; 11; 14; 15g and i = 10. V. TRADEOFF BETWEEN FAIRNESS AND REVENUE MAXIMIZATION More often than not, MMF is not the carried traffic intensity vector which yields the maximum revenue. At present, we consider the rate-per-call-per-unit time to be uniform throughout the system and hence the total carried traffic intensity in the system gives the rate at which revenue is earned, except for a multiplicative constant which we take as unity. We define the marginal revenue (see [8]) of a channel assignment algorithm A (or the corresponding carried traffic intensity A vector x ~ (r )) at load r and number of channels n, as TA (r; n) = N i=1 (b) Fig. 5. Maxmin fair call acceptances (c ) in a 19-cell system. S = 6; 9; 10; 11; 14; 15g (central cell and its neighbors). (a) Uniform traffic. (b) Nonuniform traffic. f5; xA i (r): So the marginal revenue of a channel assignment algorithm is the rate at which revenue is earned per channel using that algorithm. Let TM (r; n) be the marginal revenue of the MMF. Let TR (r; n) = N i=1 xR i (r) (a) ~ (r ) the channel assignbe the maximum marginal revenue [8] and R R ment algorithm (x ~ (r ) the corresponding carried traffic intensity vector at load r ) yielding this marginal revenue at load r . Let TM (r) and TR (r) be the corresponding asymptotic marginal revenues. Consider the following examples. Example V.1: 1) Consider the system of Example IV.1 1). TM (r) and TR (r)6 have been plotted versus r in Fig. 6(a). For r > 247=49, TM (r) < TR (r). 2) Consider the system of Example IV.1 2). For r TM (r) < TR (r) (refer to Fig. 6(b)). > 4:0, Clearly, TM (r; n)  TR (r; n) for all systems. Generally this inequality is strict. Whenever this inequality is strict, the channel assignment algorithm attaining TR (r; n) is unfair compared to that attaining TM (r; n). In other words, revenue is maximized at the expense of fairness. If the operator maximizes revenue at the cost of fairness, it may cause customer dissatisfaction (customers will be dissatisfied throughout the system as they will experience poor-quality service when they move to certain cells) and possibly customer desertion to a competitor operator, which will result in reduced r and hence reduced revenue. Instead, if the network operator had opted for a carried traffic 6Computation of T (r ) has been discussed in [7]. (b) Fig. 6. Loss in revenue brought about by max-min fairness for a 19-cell system. (a) Uniform traffic. (b) Nonuniform traffic. intensity vector which fetches less revenue but is fairer, customer desertion may have been less and it is possible that the ultimate revenue earned would be greater. While we are still in the process of modeling the effects of customer desertion on fairness and revenue, it seems plausible that the ideal channel assignment algorithm in many situations would earn less marginal revenue than TR (r; n) (which we get by not caring about fairness) and more than TM (r; n) (which we get by being max-min fair). It would also, in some sense (not necessarily that of Sec~ (r ) but not “as fair” as the MMF channel tion II), be “fairer” than R ~ (r ) at load r . A possible approach is to fix the assignment algorithm M IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 8, AUGUST 2002 ( ) ( ) revenue at some value between TM r; n and TR r; n , say m, and choose the channel assignment algorithm corresponding to the MMF among those carried traffic intensity vectors that earn a marginal revenue of at least m. We denote this carried traffic intensity vector by M m; r; n (xM m; r in the asymptotic limit). Consider the folx lowing example. ~ ( )~ ( ) Example V.2: 1) Consider the system of 19 cells with uniform traffic (refer to Example IV.1 1)). At r r = : = , TR and TM r = : . A carried traffic intensity vector xR r which maximizes the marginal revenue in the asymptotic limit (computed as per [7]) has a blocking probability of 50% in the central cell (cell 10) and 0% in all other cells. : gives following On the other hand, xM m; r at m blocking probabilities: ( ) = 37 7 = 5 2856 = 38 7 ( ) = 247 49 = 5 0408 ~ () ~ ( ) b ; : ; i ; ; ; ; ; i ; ; ; ; ; ; ; x m; r x : = r x x : ; r r ( ) = 4 348 ~ () =45 ( )= ~ ( =43 ) 15 553% = 10 2 f5 6 9 11 14 15g i = 6 668% 0% otherwise. Clearly, ~M (4 3 ) is much fairer than ~R ( ), while it fetches only slightly less revenue (4 3 as compared to 4 348) at = 4 5. Again this suggests that a network operator may be better off in the long run sacrificing the additional revenue achieved by ~R ( ) and using ~M (4 3 ) instead, at = 4 5. MMF gives a blocking probability of 11.11% in cells : b ; : i ; i ; ; ; ; ; ; x : ; r x r : r r x : ; r : 5, 6, 9, 10, 11, 14, 15, and 0% blocking in other cells. The value of m could possibly be the result of modeling customer desertion in an appropriate manner. Further investigation in this direction is an interesting topic for future research. We have so far assumed that the set of carried traffic intensity vectors F m; r; p; n which fetch a marginal revenue of at least m at r has an MMF. This can be proved using the following theorem. ( ~ ) Theorem 8: If D is any nonempty, convex, closed, bounded subset N , D has an MMF. of R+ Note that the assumption of coordinate convexity is not required. Consider a set s= D A~ 2D ( ~) S A where ( ~) = f ~ : (0 0. . . 0)  ~  ~g S A V ; ; ~ iV ;  ciV ; M i ~ ~2  is not fairer than V the MMF in Ds . M ( ( ) D i 6= F R N = ~: V n = j j: s . This contradicts the fact that M~ is D ~ ) = \ ( ), where F m; r; p; n intensity vectors at r; R m for some i < c ~ ~ ~ ~ ~ c ~ 0 R m F is the set of carried traffic (and traffic pattern p) and N i=1 V i x  m ~ where xVi is the ~ th component of V i ( ) Clearly, R m is closed and convex. F is also closed and convex. Thus, the intersection of these two is also closed and convex. Since F is bounded, F m; r; p; n  F is also bounded. m  TR r; n means xR r which fetches revenue equal to TR r; n  m is in set R m and it is also there in F . Thus, the intersection is nonempty. F m; r; p; n  F  R+NN. F m; r; p; n is a nonempty closed, bounded, convex subset of R+ . Hence, F m; r; p; n has an MMF by Theorem 8. () ( ) ( ~ ) ( ~ ) ( ) ( ( ) ~ ) ( ~ ) VI. CONCLUSION AND FUTURE WORK : : x r ~ = 2) Consider the system of 19 cells with nonuniform traffic (refer to Example IV.1 2)). At r r : and TM r : , TR : . A carried traffic intensity vector xR r which maximizes the marginal revenue in the asymptotic limit (computed as per [7]) has a blocking probability of 26.98% in the central cell (cell 10) and 0% in all other cells. On the other hand, xM m; r at m : gives following blocking probabilities: 4 25 ~ r : r ~ where we say V1  V2 if xiV  xVi , for each i 2 U . We call Ds the coordinate convex extension of D . Thus, for every vector V 2 Ds , there exists some vector A 2 D such that  ciV  ciA for all i 2 U . Clearly, Ds  D , hence Ds is coordinate convex. Ds is closed, bounded, convex, nonempty if D has these properties. (It can be shown that coordinate convex extension of a closed, N is also closed, bounded, convex, bounded, convex, nonempty set in R+ and nonempty.) Thus, by Theorem 2 Ds has an MMF. Let it be M . Clearly, M is fairer than all vectors in D  Ds . Hence, M is the MMF in D if it belongs to D . If M 62 D , M 2 Ds n D . Thus, there exists V 2 D such that = 52 11 175% 2 f5 6 9 10 11 14 15g 2 f2 4 7 13 16 18g i = 0 287% 0% otherwise. Clearly, ~M ( ) is much fairer than ~R ( ), while it fetches only slightly less revenue (5 2 as compared to 5 286) at = 38 7. This suggests that a network operator may be better off in the long run sacrificing the additional revenue achieved by ~R ( ) and using ~M (5 2 ) instead, at = 38 7. : 2425 V A We have developed the notions of relative and absolute fairness. We have obtained a simple necessary and sufficient condition for relative fairness which is useful in determining if a vector is fairer than another. We have shown that any subset of RN satisfying certain properties has an MMF and the set of carried traffic intensity vectors satisfies those properties. We have shown that the MMF is unique. We have been able to specify a fixed channel assignment algorithm which yields the MMF at all loads, in the asymptotic limit. Future research may be directed toward determining optimal channel assignment algorithms that would maximize revenue taking into account the effects of customer desertion. We would like to point out that many of our results are very general in nature and apply to a large class of other networks as well. The problem of max-min fair bandwidth allocation is very relevant in context of other networks as well, e.g., automated teller machine (ATM) networks. The set of feasible allocations often satisfies the conditions of Theorem 8. Thus, we know from Theorem 8 that the max-min fair allocation exists uniquely (uniqueness follows from Theorem 5) and the construction of Theorem 2 actually yields this allocation if the set of feasible allocations is known. Our results may find application in the problem of max-min fair allocation of available bandwidth to competing available bit rate (ABR) traffic in ATM networks with minimum cell rate requirements (MCR) [1]. 2426 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 8, AUGUST 2002 REFERENCES [1] S. P. Abraham and A. Kumar, “Algorithms for max–min fair rate control of ABR connections with nonzero MCR in ATM networks,” in Proc. Conf. Communication Technologies, 1996. [2] C. Berge, Hypergraphs. Amsterdam, The Netherlands: North-Holland, 1989. [3] D. Bertsekas and R. Gallager, Data Networks. Englewood Cliffs, NJ: Prentice-Hall, 1987. [4] E. Kreyszig, Introductory Functional Analysis With Applications. New York: Wiley, 1978. [5] F. Kelly, A. Maulloo, and D. Tan, “Rate control for communication networks: Shadow prices, proportional fairness and stability,” J. Oper. Res. Soc., vol. 49, no. 3, pp. 237–252, Mar. 1998. [6] V. H. Macdonald, “The cellular concept,” Bell Syst. Tech. 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