Capacity of Muti-service Ceuar Networs with Transmission-Rate Contro: A Queueing Anaysis Eitan Atman INRIA, BP93, 2004 Route des Lucioes, 06902 Sophia-Antipois, France aso CESIMO, Facutad de Ingeniería, Univ. de Los Andes, Mérida, Venezuea atman@sophia.inria.fr ABSTRACT In this paper we compute the upin capacity of powercontro CDMA mobie networs with an ideaized power contro, that contain best-effort type appications, i.e. appications whose transmission rate can be controed. An arriving best-effort ca is assumed to have a fixed amount of traffic to send, so the transmission rate assigned to it determines the duration of the ca. We aow for muti-services (so that mobie stations have different quaity of service requirements). Unie some previous pubished wor where soft bocing was considered (and the system was thus aowed to operate beyond capacity), we assume that a ca admission mechanism is impemented in order to prevent a new ca to arrive when the system is aready saturated. This guarantees the quaity of service of ongoing cas. Our first resut is that sowing the transmission rates in the case of a singe ce with homogeneous quaity of service characteristics increases capacity. This suggests that there is a imit capacity that can be approached when sowing down the transmission rates. We identify this imit and show that it has the foowing property: as ong as the arriva rate of information is beow some eve, bocing probabiity can become arbitrariy sma by sufficienty sowing down the transmission rates. We then extend the resuts to the genera heterogeneous and muti-ce case. Categories and Subject Descriptors G.3 [Probabiity and Statistics]: Queueing theory; C.2.1 [Computer Communication Networs]: Networ Architecutre and Design Wireess communication Genera Terms Performance, Theory Permission to mae digita or hard copies of a or part of this wor for persona or cassroom use is granted without fee provided that copies are not made or distributed for profit or commercia advantage and that copies bear this notice and the fu citation on the first page. To copy otherwise, to repubish, to post on servers or to redistribute to ists, requires prior specific permission and/or a fee. MOBICOM 02, September 23 28, 2002, Atanta, Georgia, USA. Copyright 2002 ACM 1-58113-486-X/02/0009...$5.00. Keywords Best-effort, Erang capacity, CDMA, muti-ce, muti-cass, queueing theory 1. INTRODUCTION The traditiona way to define capacity has been to as how many cas can a system hande. Various definitions have been used. The Erang capacity which has been used in teephony networs is a probabiistic definition, it specifies the arriva rate of cas that the system can aow so that the probabiity of bocing of an arriva is ower than some threshod. Another version of this definition has been introduced in [17] for the wireess context where the capacity is taen to be the rate of cas that the system can aow so that the probabiity that the quaity of service is not attained is sufficienty sma; here cas are not boced when exceeding the imit of the system to provide the required quaity of service. In the above definitions, the transmission rate used by a ca is a fixed constant which may be cass dependent. Third generation wireess networs aow for mutimedia appications and new services are proposed, in particuar fie transfers, Internet browsing and eectronic mai. These non-interactive appications are ess sensitive to the assigned throughput. We coud consider them as part of a best-effort service in which the transmission rate can be assigned by the base-station For a given rate of arriva of best-effort sessions, or cas, the capacity of the system wi depend on the assigned throughput. However, we assume that the tota voume V s of traffic created by an appication s does not depend on the assigned traffic. The duration that this session wi be present and wi occupy networ resources is V s/r(s) where R(s) is its assigned transmission rate. We assume that the power contro is such that the energy per bit of the best-effort appication s does not depend on its transmission rate. Adding this new fexibiity of transmission rate assignment, the capacity of best-effort appications can now be defined as the number of sessions that the networ can hande assuming the best possibe assignment of transmission rates. We sha aso consider the case of a system with both best-effort cas as we as Rea-Time (RT) appications whose throughput is fixed and is not controed. For simpicity, we do not tae into considerations the effects of imperfect power contro that have aready been considered in many previous papers, see e.g. [2, 17]: we assume that power contro is instantaneous and we ignore saturation
phenomena that impose in practice a maximum on the transmitted power of a mobie. We indicate however (Remar 5) that effects of non-idea power contro can be incuded in our mode by a simpe transformation of the probem s parameters. We restrict in the beginning to a singe ce for which we obtain an expicit expression for the capacity of besteffort traffic. We further study combination of best-effort with rea-time (non best-effort) ca casses. We finay discuss the extensions to the muti-ce case. The paper is structured as foows. We begin by presenting at section 2 nown concepts of capacity as we as a new concept adapted for best-effort casses. We aso show in this section that the capacity of a system with homogeneous quaity of service increases when throughputs are sowed down. We then compute the best-effort capacity at Section 3 which is reated to Erang capacity with very ow input transmission rates. We proceed in Section 4 to study a system containing both rea-time as we as best-effort ca casses. We discuss the muti-ce case in Section 5. We end with a section that presents some perspectives and concusions. 2. DEFINITION OF CAPACITIES We begin by introducing in Subsection 2.1 capacity notions for a fixed transmission rate assignment, with a fixed number of mobies of each cass. Based on that, we introduce in Subsection 2.2 an extension of the we nown Erang capacity to the muticass situation and show how it is computed. In Subsection 2.3, we study the infuence of the transmission rates of best effort connections on the Erang capacity for the case of homogeneous quaity of service. Note that owering transmission rates has two conficting impacts: on the one hand, ess resources are needed at a given time for handing the connections whose transmission rates were sowed down. On the other hand (and in contrast to rea time connections), the transmission duration of such cas becomes onger, since the amount of information to be transmitted does not depend on the transmission rate. We show that the overa resut of these competing effects is sti that sowing the transmission rates increases capacity. This suggests that there is a imit capacity that can be approached when sowing down the transmission rates. We identify this imit in Section 2.4 for the singe-cass case, and show that it has the foowing property: as ong as the arriva rate of information is beow some eve, bocing probabiity can become arbitrariy sma by sufficienty sowing down the transmission rates (this is simiar to the behavior of the we nown Shannon capacity in information theory, in which we can get an arbitrariy ow error probabiity if we use sufficienty ong codes, as ong as we transmit at a rate beow the capacity). The muti-cass case is then discussed in the foowing section. 2.1 The case of a fixed number of mobies Consider upin power contro of a muti-service CDMA system. Consider an arbitrary sector within some arbitrary ce. We consider a set K {1,..., } of best-effort service casses. Exampe of best-effort appications are fie transfers, voice mais, fax. Let M(s) be the number of ongoing cas of cass s which are active, and et M (M(1),..., M()) be the vector of number of active mobies. We assume that when a fixed vector M is given, the foowing standard equation [9] is used to determine the power P (s) that shoud be received at the base station from mobie s K P (s) N + I own + I other P (s) (s), s 1,..., (1) where N is bacground noise, I own is the tota power received from mobies within the considered sector, and I other is the tota power received from mobies within other sectors and other ces. (s) is the target ratio of the received power from mobie of cass s to the tota interference energy received at the base station, and is given by (s) E(s) W N o R(s). Here, E(s) is the energy corresponding to a transmitted bit of type s, N o is the therma noise density, W is the spreadspectrum bandwidth and R(s) is the transmission rate (in bits/s) of cass s service. Remar 1. Note that we impicity assume that the target vaue (E(s)/N o) does not depend on the transmission rate R(s). This is a standard assumption in the iterature, see e.g. [7, 8]. In practice, however, it may depend on R(s), see e.g. [4, p. 151, 222, 239]. But as we see from [4, Fig. 10.4, p. 222], it is cose to a constant throughout ong range of bit rates. For exampe, between 16Kbps and 256Kbps, the maximum variation around the median vaue is ess than 20%. We thus propose to tae for the vaue of (E(s)/N o) its average or median vaue over the range of interest. However, if the exact dependence is avaiabe anayticay, it can be incuded into our mode. Remar 2. Note that the required quaity of service refected through (s) is not ony a function of the appication but aso depends on the transport ayer. For exampe, to achieve a reiabe fie transfer, typicay TCP/IP protoco is used at the transport ayer which can support pacet osses (due to errors or to congestion) of a few percent by resorting to retransmission of ost pacets. The same appication may thus be transmitted using different eves of power per bit (which woud correspond to different casses in our modeing) depending on the transport protocos used. We have I own M(j)P (j). (2) To mode inter-ce interference we mae the standard simpifying assumption [9] that I other i I own (3) for some given constant i which is obtained from measurements. We sha find it more usefu to rewrite (1) as where (s) P (s) N + I own + I other (s), (4) (s) 1 + (s) (s) (s) 1 (s),
s 1,...,. Soving the set of equations (4) yieds N (s) P (s). (5) 1 (1 + i) M(j) (j) (The soution is in particuar simpe to obtain, since by mutipying the eft side of (4) by M(s) and summing over s we get a singe equation with the singe unnown M(j)P (j). This then provides immediatey the vaues of a the P (s).) The poe capacity of the system can be defined as the poyhedron M of vectors M that mae the denominator of (5) vanish. It is thus given by M {M : 1 (1 + i) M(j) (j)}. We say that M 1 < M 2 (in the Pareto sense) if M 1(j) M 2(j) for a j 1,...,, with strict inequaity for at east one j. It is easiy seen that the soution P (s) of (5) is finite if and ony if M < m for some m M. We may sighty change the above definition so as to tae into account that M(j) are in practice integer numbers (M beongs to ). Definition 1. Let M be the finite subset of for which 1 > (1 + i) M(j) (j), and et η max (1 + i) m M m(j) (j). (6) We define the Integer Capacity M B of the system as the boundary of M for which any additiona ca woud resut in an infinite power assignment in (5), or equivaenty the set of M s for which η (1 + i) M(j) (j). Remar 3. We note that the power corresponding to one service is finite in (5) if and ony if it is finite for a services. Thus if a ca admission contro is not used to avoid exceeding the Integer Capacity of the system then the resut is harmfu not ony for the ca accepted beyond capacity but aso for a other ongoing cas. We concude with another usefu definition. Definition 2. The bocing set M j B of cass j K is defined as the subset of M for which another ca from cass j cannot be accepted, i.e. m M j B if and ony if m M and m + e j / M, where e j is the unit vector in direction j. 2.2 Random number of mobies: a singe ce We consider the case of a singe isoated ce consisting of a singe sector (i.e. i 0). We now mae standard statistica assumptions [17]: cas of cass s arrive according to a Poisson process with intensity λ(s), and their duration is exponentiay distributed with parameter µ(s). Denote λ λ(s). Let ρ(s) : λ(s)/µ(s) be the oad of cass s cas. At this point we assume that a cas are aways active. Remar 4. If this were not the case and a ca of cass s were active with probabiity p s, our mode coud sti be usefu if we focus on the process of arriva of active periods of cas, and assume that it can be modeed as an M/M/S/S system. Note that in that case, the bocing events wi correspond to bocing of an active period rather than of the whoe session. Aternativey, one may introduce an activity factor; we mention a method for transforming a probem with activity factor to ours in Remar 5. An exampe of a best-effort appication with activity and inactivity periods is the HTTP1.1 [10]. We note that the vector of number of cas is an irreducibe ergodic finite Marov chain whose state space is given by M. Let π ρ(m) denote the steady state probabiity of this Marov chain. Extending the definition of Erang capacity to our mutiservice case we have: Definition 3. Define the Erang capacity EC(ɛ) as the set of vectors ρ (ρ(1),..., ρ()) such that the corresponding bocing probabiity P B(ρ) is smaer than a given ɛ. The Erang capacity is thus a set instead of a singe constant. Theorem 1. The steady state probabiities of the Marov chain are given by π ρ(m) 1 G ρ where G ρ ρ(s) M(s), M M, M(s)! m M ρ(s) m(s). m(s)! The probabiity P s B(ρ) that an arriving ca of cass s is boced and the average goba bocing probabiity P B(ρ) are given by P s B(ρ) m M s B π ρ(m), P B(ρ) λ(s) λ P s B. Proof. The statements of the theorem are very simiar to [8] and references therein (see aso [11]) and the proof can thus directy be obtained from those references. For competeness we setch the basic idea. The Marov chain has the same structure as that used in muticass oss systems [6, 14] without power contro, in which there is a set K of casses of cas with the same distribution of arrivas and durations of cas as in our case, in which there is a tota given bandwidth of η units, and in which cass i K requires an amount (i) of bandwidth. We thus interpret the target ratios (i) as bandwidth requirement in the equivaent oss mode and obtain the steady state probabiities. Since the arriva processes are Poisson, the distribution upon arriva equas to the steady state distribution (the so caed PASTA property, see e.g. [18]), which provides the formuas for the bocing probabiities. Remar 5. In [8] (and references therein) it is shown in fact that the genera form of the steady state probabiities in Theorem 1 extends aso to the case of non-perfect power contro and to the case where the transmission of a mobie (7)
may be interrupted by sience periods. To hande the atter case in our framewor, we may proceed as in [8] (see aso [17]) and introduce the activity factor α(s) (between 0 and 1). Then M(j) in (2) is repaced by α(j)m(j). The impact on the capacity woud be to mutipy in (6) each m(j) by α(j). We see from the capacity definition (Definition 1) that the capacity (and hence bocing probabiities) of a system with activity factors α(j), j 1,...,, is equivaent to the capacity of the origina mode we anayze (without activity factors) but where each (j) is repaced by α(j) (j). Aso, if we use the approach of [8] we may incorporate the imperfect power contro in our mode by further mutipying the (j) s by some constants that depend on the standard deviation of the received signa to interference ratio. We concude that in spite of the simpicity of our mode, it can be used in fact to study aso non-perfect power contro and non constant transmission rates. A more precise way to incude both the activity factor and imperfect power contro can be done as in [17] but it does not ead to anaytica expressions. 2.3 The case of homogeneous quaity of service In order to iustrate the impact of transmission rates on capacity, we introduce the specia case in which a casses have a common vaue of (s), s 1,...,. The arriva process of cas of cass s is Poisson with parameter λ(s) as before, yet the ca duration distribution of a cass s ca may have a genera distribution G s with mean µ(s) 1. The system evoution is equivaent to that of a singe cass with arriva rate of λ λ(j) and where the distribution of the duration of an arriva, G, is according to G s with probabiity λ(s). Thus the expected duration of the arriva is µ 1 λ(j) λ µ(j) 1. We can thus define the oad of the system as ρ λ µ ρ(j). The integer capacity of the system is given by M B max{m N : m < 1} 1 1 (8) where x is the smaest integer greater than or equa to x. Due to the genera distribution of the duration of cas, the process of number of cas in the system, taing vaues in M {1,..., M B} is no more a Marov chain. Sti we have the foowing: Theorem 2. The steady state probabiities of the Marov chain are given by π ρ(m) ρ M /M! m M ρm /m!, M M. The probabiity that an arriving ca of cass s is boced does not depend on s and is given by P B(ρ) π ρ(m B). requires an amount of one unit bandwidth. In this equivaent system, the steady state distribution is nown to be insensitive to the ca duration distribution (it ony depends on its expectation), see [15]. Since the arriva processes are Poisson, the distribution upon arriva equas to the steady state distribution (the so caed PASTA property, see e.g. [18]), which provides the formuas for the bocing probabiities. Remar 6. One can even further reax the statistica assumptions under which Theorem 2 hods, by aowing the ca durations to be genera stationary ergodic. This means that the duration of successive cas need not be independent. The Theorem then foows from the insensitivity resut of [3]. Dependence between durations of successive cas may be usefu especiay when a ca represents in fact an active period rather than a whoe connection. For exampe, a singe HTML appication may contain severa successive fie transfers whose durations may be correated. We sha assume beow that 1 (or equivaenty 1 ) is an integer. We show the impact of the assigned throughput. Assume that a casses are best-effort. Keeping the same vaue E(s) of energy per bit for a casses, we sow the transmission rate of a casses by dividing them by a constant a > 1. In other words, we repace R(s) by R a (s) R(s)/a. As a consequence as we as the ca durations are divided by a. Hence with the sower transmission rates we get a new oad ρ a ρa. From (8) it then foows that the new integer capacity is M B,a a/ 1. Note that for integer vaues of 1 and a we have M B,a a 1 a + 1 ã amb. (9) The steady state probabiities for the new system are π ρ,a(m) (aρ) M /M! M B,a, M 0,..., MB,a. m1 (aρ)m /m! Next we show the infuence of a on the bocing probabiity and the Erang capacity. Theorem 3. As a increases, the bocing probabiity decreases. Proof. Let X be the state (goba number of cas) of the initia system and Y the state of the new one. Define Z max(0, Y M B,a + M B). Thus Z taes vaues in {0, 1,..., M B}. For a random variabe V defined on M we define r V (0) 0, and r V (m) We have for m > 0 r X(m) m ρ, P (V m 1), m 1,..., M B. P (V m) Proof. Simiar to the proof of Theorem 1. The process of number of cas has the same structure as that used in oss systems [6, 14] without power contro, in which there is a tota given bandwidth of M B and in which each ca r Z(m) m + MB,a MB aρ m + (a 1)/ aρ m + a/ 1 aρ
Thus r X r Z(m) (a 1)(m 1 ) aρ 0. It then foows (see [13] and references therein) that Z st X or equivaenty E[f(Z)] E[f(X)] for any nondecreasing function f. In particuar, by taing f to be the indicator f(x) 1{x M B,a} we concude that P (X M B) P (Z M B) P (Y M B,a) from which we concude that the bocing at the origina system indeed has greater probabiity than in the system a. If we consider the homogeneous case as an equivaent system with a singe cass (as in the first paragraph of the Subsection), then the fact that the bocing probabiity decreases with a impies that the Erang capacity of the onedimensiona system increases (we use the fact that for fixed a, the bocing probabiity increases with the input rate, see [13, 12]). 2.4 Best-effort Capacity We now define the capacity in systems with best-effort appications. To motivate our definition, we go bac to the homogeneous mode of Subsection 2.3. We showed there that the bocing probabiity decreases as the transmission rate decreases. We now show through a simpe cacuation that if ρ < M B then the bocing probabiity tends to zero as a tends to infinity. This wi then motivate us to introduce a definition of capacity which, in contrast to the Erang capacity, does not depend on a given parameter ɛ. It wi be more reated to the Shannon capacity concept. Theorem 4. Consider the homogeneous system introduced in Subsection 2.3, and et ρ < M B. Then the bocing probabiity tends to zero as a tends to infinity. Proof. Choose any ɛ > 0, and choose Denote m max(1/ɛ, M B). n 0(ɛ) m/(m B ρ). We have for any integer a > n 0. /M B,a! PB a ρm B,a a M B,a j0 ρ j a/j! (aρ)m B,a /(M B,a)! M B,a jm B,a m (aρ)j /j! M B,a(M B,a 1)(M B,a 2) (M B,a m) (aρ) m + + + MB,a(MB,a 1)(MB,a 2) (aρ) 3 MB,a(MB,a 1) (aρ) 2 + MB,a aρ + 1 1 M B(M B 1 a )(MB 2 a ) (MB m a ) ρ m + + MB(MB 1 a )(MB 2 a ) ρ 3 + MB(MB 1 a ) ρ 2 + MB ρ + 1 1 < 1 m ɛ. The ast equaity foows from (9). The inequaity before the ast foows since for any 0 j m and a > n 0(ɛ) we have M B j/a ρ. This estabishes the proof. Remar 7. Note that the above proof shows that for any given ɛ, if we sow down the transmission rates by a factor arger than n 0(ɛ) then the bocing probabiity is smaer than the given ɛ. An aternative simper proof that does not give a bound on rate of convergence is as foows. Let Y (a) be a Poisson random variabe with parameter aρ. Then PB a P (Y (a) MB,a) P (Y (a) M B,a) P Y (a)/a M B P Y (a)/a M B (10) (where we use (9)). Let Y s, s 1,..., a be i.i.d. Poisson random variabes with parameter ρ. Then Y (a) has the same a distribution as Ys. Due to the strong aw of Large numbers, Y (a)/a converges P -amost surey to its expectation, ρ, as a. Since ρ < M B, this impies that the enumerator of (10) converges to zero and the denominator to 1 as a, which estabishes the proof. In order to define the best-effort capacity, we need some more definitions. Consider a system where a casses are best-effort casses, where the amount of data that a ca of cass s has to transmit has an exponentiay distributed size with parameter ζ(s) (its expected size is 1/ζ(s)) and the arriva rates of cas of the casses are given by the vector λ (λ(1),..., λ()). Let the assigned transmission rate of cass s cas be R(s), and denote R (R(1),..., R()). Then the transmission time of cass s is an exponentiay distributed random variabe with parameter µ(s) R(s)ζ(s) (the expected ca duration is thus (ζ(s)r(s)) 1 ). Define the utiization density ν(s) λ(s)/ζ(s), and define the vector ν (ν(1),..., ν()). Define so that δ(s) E(s) W N o (s) R(s)δ(s) and (s) and et δ (δ(1),..., δ()). R(s)δ(s) 1 + R(s)δ(s), Definition 4. Consider the system described above with a given δ. Define the BE (best-effort) capacity as the supremum of the set of vectors ν (ν(1),..., ν()) for which for any ɛ > 0, there exists a vector R (R(1),..., R()) of transmission rates such that P s B(ν, R) < ɛ for a s 1,...,. 1 Numerica Exampe: Consider a singe cass homogeneous system as described in Section 2.3, that handes best-effort sources and offers them a high-speed connection, i.e. a arge transmission rate of R 160KB/sec (i.e. 1.28 Mbps). Let 0.199 so that the system is dimensioned such that its integer capacity is 5, i.e. no more than 5 cas can be simutaneousy handed. Assume 1 The supremum is taen in the Pareto sense, i.e. a vector n beongs to a supremum set satisfying a property, if there is no other vector arger (in the Pareto sense) than n which satisfies that property, and if for any ɛ > 0 there exists a vector ν satisfying the property such that ν(s) n(s) ɛ for a s 1,...,.
that the average amount of information (e.g. the average fie size) of a connection is 10KB (the average size of fies transfered on the Internet is nown to be between 8-12 KB, see [16] and references therein). The average duration needed 10KB 160KB/sec for handing a session is µ 1 62.4msec, and µ 16.03. Assume that we wish that the bocing probabiity be inferior to 1%. Using Theorem 2 we see that the Erang capacity of the system is ρ 1.361, which means that for having at most 1% of osses the rate of arriva of sessions shoud be imited to λ ρµ 21.8 cas per second. Tabe 1 shows the gain by sowing the transmission rates by a factor of a. For each a it gives the Erang Capacity EC(1%) as we as the rate λ of arriving cas that the system can hande without exceeding 1% of bocing. In particuar, we see that we doube the capacity by sowing the transmission rates by a factor of around five. This coud indicate that among connections that have the same voume of information to transmit, connections that are five time sower use haf of the effective amount of resources than the others. Hence if we were to assign prices per voume of information transmitted as we as of the speed transmitted, the sower connections coud be priced the haf per transmitted voume than the origina ones. The Best-effort capacity for this probem is ρ 5. 3. COMPUTING THE BEST EFFORT CA- PACITY Focusing on the singe cass, we showed in the previous Section that as ong as the arriva rate of information is beow some eve, bocing probabiity can become arbitrariy sma by sufficienty sowing down the transmission rates. In this section we estabish the same resut for the muti-cass case. Theorem 5. The BE Capacity of the system (with Poisson arrivas with rate vector λ and with sizes of cas exponentiay distributed with (vector) parameter ζ) is given by the set of ν satisfying The proof is based on two parts. ν(s)δ(s) 1. (11) Lemma 1. If ν satisfies ν(s)δ(s) > 1 then the bocing probabiities PB(R) s for casses s 1,..., satisfy for any R s P s Bδ(s)ν(s) > s δ(s)ν(s) 1 > 0. Proof. Assume that for some ν, ν(s)δ(s) > 1. Choose an arbitrary R. Define the foowing random processes, taen to be right continuous with eft imits: M t(s): the number of s-type cas at time t, A t(s): the number of arrivas of s-type cas ti time t, D t(s): the number of s-type cas that ended ti time t, B t(s): the number of s-type cas that have been boced by time t. Then we have M t(s) M 0(s) + A t(s) D t(s) B t(s). (12) Let µ(s) R(s)ζ(s). Note that at time t, the departure rate of cass s has a stochastic intensity of M t(s)µ(s). The oss rate of cass s cas at time t is given by λ(s)p s B(t) where P s B(t) is the probabiity that an arriving ca of type s is boced at time t, i.e. that M t M s B. Differentiating (12) and taing expectations we thus obtain de[m t(s)] λ(s) µ(s)e[m t(s)] λ(s)p s dt B(t). (13) The process M t being a finite irreducibe Marov chain, converges to a steady state distribution, for which the expected time derivative vanishes above. Mutipying (13) by δ(s)/ζ(s), taing the sum over the casses and omitting t from the notation (to indicate that we are at steady state), we obtain (s)e[m(s)] PBδ(s)ν(s) s δ(s)ν(s) > δ(s)ν(s) 1 which estabishes the proof. (We used the fact that δ(s)µ(s) ζ(s) (s) > (s) and that s (s)m(s) cannot exceed 1.) The above Lemma gives a ower bound on the bocing probabiity when ν(s)δ(s) > 1. For the homogeneous case this gives, in particuar, P B > δ(s)ν(s) 1 δ(s)ν(s). Next, we mae the foowing observation on the way (s) scaes with a. We have for a 1, (s) a a(s) (s) a + (s) 1 + 1 1 + 1 1 a (s) (s) a 1 (s) a. (14) Lemma 2. Consider the system with fixed vectors δ and ν, and with an arbitrary given rate vector R with positive components. Assume that ν(s)δ(s) < 1. Then the imit as a of the bocing probabiity when R is divided by a is zero. Proof. Let Y s be independent Poisson random variabes with parameters ρ(s), s 1,...,, and et Z (j)yj. Then the steady state probabiities (7) can be rewritten as P (Ys Ms) π ρ(m) M M, P (Z η) where η is given in (6). Then the bocing probabiity satisfies P s B(ρ) P (Z 1 (s)) P (Z < 1) P (Z 1 (s)). (15) 1 P (Z 1) Next we use Chebycheff bound for Z; for any positive rea numbers α and β we have: P (Z α) E[exp(βZ)]. (16) exp(βα)
Sowing factor a 1 2 3 4 5 6 20 0.199 0.110 0.0764 0.0583 0.0473 0.0398 0.0122 0.248 0.124 0.0827 0.0620 0.0497 0.0414 0.0124 M B,a 5 9 13 17 21 25 81 Erang Cap. EC(1%) 1.361 1.891 2.202 2.413 2.568 2.688 3.315 Arriva rate λ 21.8 30.3 35.29 38.67 41.15 43.08 53.13 Average ca duration in msec 62.4 124.8 187.2 249.6 312 374 1248 Gain in % 0 43.7 67.4 83.4 95 104 144 Tabe 1: Gain in Erang capacity by sowing transmission rates by a factor of a We reca that the PGF of the Poisson random variabe Y s is given by E[ξ Ys ] exp(ρ(s)(ξ 1)). It then foows that E[exp(βZ)] E exp β (s)y s exp ρ(s)(exp[β (s)] 1). Choose an arbitrariy sma ɛ > 0 and et β > 0 be such that Then we obtain for (16): Thus exp[β (s)] 1 β (s)(1 + ɛ). P (Z α) exp β (1 + ɛ) P B(ρ) ρ(s) (s) α. exp β (1 + ɛ) ρ(s) (s) 1 + (s) 1 exp β (1 + ɛ) ρ(s) (s) 1 (17) We now divide a transmission rates by a > 1. Then the new bound derived from (17) for the bocing probabiities is obtained by defining Z a (j)yj(a) where Y j(a) have Poisson distributions with parameters aρ(s). We obtain P s B,a(ρ) exp β a{(1 + ɛ) aρ(s) a(s) 1} + a(s) 1 exp βa (1 + ɛ) aρ(s) a(s) 1 exp β a{(1 + ɛ) ρ(s) (s) 1} + (s) 1 exp βa (1 + ɛ) ρ(s) (s) (18) 1 where the ast inequaity foows from (14). Condition (11) impies that (1 + ɛ) ρ(s) (s) 1 < 0 for a ɛ sufficienty sma, which impies that P s B,a tends to zero as a. Remar 8. Note that the above proof provides aso a bound on the rate of convergence of the BE to 0 as a. A more direct proof that does not provide a rate of convergence can be proposed by extending the approach in Remar 7. 4. COMBINED REAL-TIME AND BEST EF- FORT APPLICATIONS In this section we examine the situation in which we can sow down the transmission time of ony some casses (of best-effort traffic); other casses that may correspond to rea time appications transmit at a fixed rate 2. More precisey, we consider BE casses enumerated by 1,..., whose transmission we may sow down and a set of rea-time casses: + 1,..., with fixed transmission rate. For each parameter a one can use (7) for computing bocing probabiities and use it then to compute the capacity. However, pursuing our approach from previous sections we proceed to compute the imiting behavior of the system as the throughput assigned to best-effort casses is sowed down. We sha show that sowing the throughputs of besteffort traffic improves their performance. Unie the case of best-effort traffic ony, we cannot expect the goba bocing probabiities to vanish as transmission rates of best-effort traffic are sowed down for any vaues of δ(s) and ζ(s), as ong as there is positive probabiity of arrivas of rea-time traffic. Yet we sha show that for a given set of parameters of best-effort cas, their bocing probabiities can be made arbitrariy sma by sowing sufficienty their transmission rate. Theorem 6. If ν(j)δ(j) 1 then at steady state, a RT cas are boced, and the system is thus equivaent to one with no RT traffic. (1) Assume that ν(j)δ(j) < 1. Define M (m( + 1),..., m()) : ν(j)δ(j) + j+1 M s B (m( + 1),..., m()) M : ν(j)δ(j) + j+1 (j)m(j) < 1, (j)m(j) 1 (s). 2 In practice aso video and voice appications may be transmitted with a ower throughput using various compression mechanisms. We do not treat this possibiity here, we aready assume that if different possibe throughputs are avaiabe for the rea-time traffic then the ones used correspond to the required quaity of these appications (of course arger compression rates resut in ower quaity). Note that sowing the throughput of rea-time appications by a factor of a does not resut in a onger ca duration (uness the ca has such a bad quaity that speaers have to repeat entire phrases. This is not an interesting case from a system design point of view).
(2) Assume that for a s + 1,..., and for a m M s B we have ν(j)δ(j) + j+1 (j)m(j) > 1 (s). (19) Then the imiting steady-state and bocing probabiity of a rea-time cass are given by Theorem 1 where M is repaced by M, M s B by M s B and where we restrict summations to casses + 1,...,. The imiting bocing probabiities of a BE casses are zero as a. Proof. Let Y s be independent Poisson random variabes with parameters aρ(s) for s 1, and with parameter ρ(s), s + 1,... Denote Z(a) a(s)y s + s+1 (s)y s. Then the bocing probabiity PB,a s of a RT cass s when transmission rates of BE cas are sowed down by an integer a > 1 is given by P s B,a P (1 (s) Z(a) < 1). (20) P (Z(a) < 1) Define Ỹ s r to be independent Poisson variabes with parameter ρ(s), r 1,..., a. Then Y s has the same distribution a as r1 Y s r, s 1,...,. Now, the strong aw of arge numbers impies that a r1 Y s r /a converges in distribution to the constant ρ(s), s 1,...,. Since im a a a(s)/ (s) 1, Z(a) converges weay to Z as a, where Define the sets Z : (20) impies that δ(s)ν(s) + s+1 (s)y s. A 1 [1 (s), 1), A 2 [0, 1). P (Z 1) P (Z 1 (s)) 0. Moreover, P (Z 0) 0. It then foows by Portmanteau s Theorem [1, p. 11], im P (Z(a) A1) P (Z A1), a im P (Z(a) A2) P (Z A2). a Then the enumerator of (20) converges to P (Z A 1) and the denominator of (20) converges to P (Z A 2). Thus, defining the event we have A : 1 im P B,a s a δ(j)ν(j) (s) < 1 δ(j)ν(j)) j+1 (j)y j P (A) δ(j)ν(j) P j+1 (j)yj < 1 P (Y M s B) P (Y M). The ast expression coincides with bocing probabiities of equivaent oss systems from [6, 14] with the casses that correspond to RT traffic. Thus this expression can be identified with the probabiities stated in the theorem. We see that if ν(s)δ(s) 1 then a rea-traffic is boced at steady-state. The system is thus equivaent to one with no RT traffic, and we can use Lemma 1 to show that BE traffic wi suffer a positive oss rate. Assume now that ν(s)δ(s) < 1. There is some ɛ > 0 such that im P (A3) 0 a where A 3 is the event given by (m +1,..., m ) : s+1 (s)m(s) > 1 + ɛ ν(s)δ(s). Hence by taing arge a, the probabiity of A 3 can be made arbitrariy sma. Moreover, due to the strong aw of arge numbers, P aso ɛ ( (s)/a)m s > ν(s)δ(s) + can be made arbitrariy sma. Hence the bocing probabiity of BE converges indeed to 0 as a since for BE traffic to be boced at the system a we need to have a(s)m s + s+1 and since im a a a(s)/ (s) 1. (s)m s 1 a(s), s 1,...,, 5. THE MULTI-CELL CASE In this section we introduce a method for approximating the bocing probabiities (and thus obtaining the Erang capacity) for fixed transmission rates for the muti-ce case. Our method is based on a mean fied approach and on fixed point arguments; we study in particuar the existence and uniqueness of the fixed point. We then present the corresponding BE capacity. 5.1 Bocing probabiity for the muti-ce case Our approach is inspired by the approximation used for computing the poe capacity [9] for the muti-ce case, which we aready mentioned at (5). We assume a symmetric system of ces. In our stochastic framewor of Poisson arrivas of cas and exponentia ca duration, it is not reasonabe to expect that (3) hods at each moment. Instead, we assume that it hods in expectation: E[I other ] i E[I own]. The mean fied approximation amounts on further assuming that the instantaneous interference from other ces is repaced by its average: I other i E[I own]. We then get instead of (5) the reation for the (random) power of type s ca: N (s) P (s) 1 M(j) (j) Q (21) where Q i E[M(j)] (j) (the randomness comes since here, M is a random variabe). For each fixed vaue of Q (possiby different than the vaue E[M(j)] (j)), we can obtain the probabiity distribution of M(s), s 1,..., (under the assumption that cas i
of cass s are boced whenever the denominator of (21) woud vanish or become negative if the ca were accepted). More precisey, define M(q) as the set of M for which the assigned power according to (21) (with a genera parameter q repacing Q) is finite, and et M s B(q) be the bocing set of cass s, i.e. M(q) (m(1),..., m()) : M s B(q) (m(1),..., m()) M : (j)m(j) < 1 q, (j)m(j) 1 q (s). Using the same arguments as those used to derive Theorem 1, we concude that the steady state probabiities π ρ(m, q) (for the given parameter q) of M is given by (7), where M is repaced by M(q); the bocing probabiities are aso obtained as in Theorem 1. Denote by E q the expectation operator that corresponds to the probabiity measure π ρ(m, q). Define F (q) i E q[m(j)] (j). We can characterize Q as the soution of the fixed point equation: q F (q). (22) Note that F (q) is in fact piecewize constant in q, and has thus discontinuities. This impies that (22) need not have a soution. However, the set of vaues of i for which a soution to (22) does not exist has Lebesgue measure zero. In other words, a sight change in the vaue of i wi yied a soution. F (q) is nonincreasing in q which impies uniqueness of the soution to (22). Indeed, et X(q) in system q. Define r q(0) 0, and (s)m(s) r q(m) P q(x m 1)/P q(x m), m M(q). The for q1 < q2 we have r q1(m) r q2(m) for a m M(q2). It then foows from the point 1 after Theorem 3 in [13] that E q1[x] E q2[x] which estabishes the monotonicity. 5.2 Best-effort capacity for the muti-ce case Using the above approach, one can now estabish the foowing using simiar steps as in Section 3: Theorem 7. The BE Capacity of the muti-ce system (with Poisson arrivas with rate vector λ and with sizes of cas exponentiay distributed with (vector) parameter ζ) is given by the set of ν satisfying (1 + i) ν(s)δ(s) 1. (23) 6. CONCLUDING COMMENTS AND PER- SPECTIVES We have studied in this paper the capacity of CDMA systems that handes best-effort traffic whose transmission rate can be determined by the networ. We assumed perfect power contro which aowed us to obtain expicit expressions for the capacity of the networ. It was shown that capacity is in fact approached by sowing transmission rates of best-effort traffic. We indicated how non-idea power contro can be integrated into our mode. In practice, however, cose oop power contro is typicay not impemented for pacet transmissions. One reason for that is that the time it taes to transmit a pacet may be too short for a feedbac contro to converge, given that cose oop power contro is updated around 1500 times per second. However, our findings suggest that the system capacity can be improved by sowing transmission rates. This woud mae the transmission duration of best-effort traffic onger, which might mae cosedoop power contro more appropriate. Sti, one coud possiby restrict our approach to those best-effort appications that are sufficienty ong such as fax, ong fie transfers (in particuar video on demand in which a whoe video fie is transferred), voice-mai, etc. Throughout our paper, best-effort cas of a given cass were assumed to have pre-determined transmission rates (and the question was how to determine them). For arge fie transfer appications, as mentioned in the previous paragraph, this modeing assumption is quite reaistic. Some best effort appications do not have a constant transmission rate, see e.g. http transfers (which contain sience periods). If the instantaneous transmission rates do not depend on the state of the system, the anaysis coud be handed within our mode using a proper transformation, as mentioned in Remar 5. Yet, in other networing contexts, one further aows the instantaneous transmission rates of best-effort appications to depend on the system s state, see for exampe the ABR (Avaiabe Bit Rate) cass in ATM networs, or the TCP congestion contro in the Internet. We coud aso consider this additiona feature in wireess networs offering integrated services, in order to better use the resources: at ow congestion periods we coud aow for arger throughputs of best-effort casses which woud reduce the duration of such cas. Even within the duration of a ca one coud consider varying the throughput (especiay to avoid dropping of cas). We sha pursue these research directions in the future. We foowed the standard modeing assumption on the arrivas of sessions (see aso [11, 8, 17]), assuming that they foow Poisson processes. This impicity impies that we have an infinite source of connections. An aternative modeing assumption can be to assume a finite popuation of sources of connections, which woud give rise to different bocing probabiities and different expressions for the capacity. The BE capacity defined here was estabished by a scaing of the system in which transmission rates were sowed down by a factor a, and consequenty, the energy per bit was unchanged, Consequenty, the integer-capacity of the system (as opposed to the Erang capacity) grew ineary in a. Arriva rate of cas, as we as the amount of information to be transmitted were not affected by this scaing. We shoud mention that an equivaent scaing has been studied in the context of oss systems (without power contro) in which the ca durations were not changed, the capacity increased by a factor a as we as the arriva rates, see [6, 5]. By taing the imit as a grows to infinity, the trajectories of the system has been shown in [5] to converge to some fuid mode.
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