METASTABILITY OF CDMA CELLULAR SYSTEMS NELSON ANTUNES, CHRISTINE FRICKER, PHILIPPE ROBERT, AND DANIELLE TIBI
Abstract. In this paper, it is shown that the coexistence of a variety of different traffics in third generation cellular networks may lead to a very undesirable behavior of the whole network: a metastability property. When this property holds, the state of the network fluctuates on a very long time scale between different set of states. These long oscillations of the network make impossible to predict the average performances of some of the key characteristics of the connections, such as the handoff blocking rate or the probability of call blocking. As a consequence, the quality of service provided by such a network can be guaranteed only by, sometimes poor, lower bounds. Experiments of networks with this behavior are presented and the analysis of a corresponding simplified mathematical model is developped. The practical implications in the design of radio resource management for CDMA cellular networks are discussed.
1. Introduction Third generation cellular networks such as the Universal Mobile Telecommunications System (UMTS), offer a wide variety of services to users, including real time audio/video applications in addition to data services such as file transfer, web browsing, . . . Naturally, these types of services have very different requirements in terms of transfer delay and loss, data transmission being very sensitive to packet loss but relatively tolerant to delay whereas real time services have strict transfer delay constraints. In CDMA-based cellular systems capacity depends on intra- and inter-cell interference, path-loss, fast fading, and shadowing, which result in time-varying cell capacity, see Lee [12]. Under these conditions it is a real challenge to guarantee the QoS of the different services. A key role is played by call admission control (CAC) policies that block new calls whenever it is necessary to preserve the network efficiency in overload. However, situations may arise, due to the random environment of the radio conditions and the mobility of users, where it is preferable also to block handoff calls rather than reducing the QoS of ongoing calls. Several studies in the design and evaluation of CAC for CDMA networks under different schemes have appeared in the literature (see, e.g., Kim et al. [11], Jeon and Jeong [10] and Liu et al. [14] and the references therein). Among them, SIR-based CAC are very important since they have a direct impact on BER (bit error rate). Long oscillations. This paper considers the case when there is no priority of service for the different classes of calls and the policy is a very simple admission control policy SIR-based: A new call or handoff call is accepted if energy-per-bit-to-total-interference density ratio (Eb /I0 ) requirements of both the existing calls and the (new or handoff) call can be met. In this context it is shown that a very intriguing phenomenon identified as bistability (metastability in general) may hold. When there are at least two classes of calls, G (greedy) and S (small) say, with different transmission rates, QoS requirements and mobility patterns, this phenomenon can be roughly described as follows. For simplicity it is assumed that a G call requires a significant fraction of the varying capacity of the cell whereas an S call needs only a small portion of this capacity. The state of the network is defined as the vector of the numbers of G and S calls in each cell. Under some conditions on the parameters, the state of the network lives for a very long duration of time around one set of states where G calls are rejected with high probability. By “very long”, one means of the order of exp(αN ) for some constant α > 0 if the region covered by the cellular network has N cells. See Bovier [3] for related phenomena in statistical physics. During this period, the network accommodates, mostly, S calls. After this long phase, due to some rare events, the state of the network moves into a second set of states where a significant fraction of G calls is accepted. And again, this phase is long before it moves back to the first region, and so on. 1
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N. ANTUNES, C. FRICKER, PH. ROBERT, AND D. TIBI
The Problem of QoS. It must be stressed that, though it is quite common that the state of networks oscillates between several classes of states, it is very uncommon that its sojourn time in each of them is very long. In this situation average performance measures, such as outage probability, new and handoff call blocking, forced termination or total average bit rate are not meaningful. Indeed, each of these characteristics is generally estimated as a time average over a reasonable time window but, due to very long sojourn times, this estimation will probably take place while the network stays around some particular set of states. Much later, the same estimation will, very likely, lead to a different value provided that the set of states has changed. Because of this phenomenon of long time scale, the quality of service experienced by a user depends on the set of states where the network is when the call is established, and not on a general average on all the possible states. This phenomenon is clearly very undesirable since, in this situation, the operator in charge of such a network is not able to guarantee some level of quality of service, except perhaps through rough lower bounds.
S0
S2 S1 Figure 1. Energy levels of φ for a network with two stable points More formally, metastability can be represented in the following way: To such a network is associated an energy function φ, see Section 5. This function defines an energy landscape in which the state of the network tends to move downward, i.e. toward the region of a local minimum of this function, like S1 or S2 in Figure 1. Since the network is subject to some random fluctuations due to call arrivals, departures or handoffs, its state will fluctuate. But, because of the stability of the local minimum, these fluctuations will have a very limited impact as long as they are reasonably sized so that the state of the network will stay around this local minimum. Only a rare event, and consequently after a very long time, will cause a large motion out of the region of this local minimum and possibly in the domain of attraction of another stable point. Related Phenomena. Up to our knowledge, it has not been realized that such a phenomenon could affect the behavior of cellular wireless networks. We claim that the coexistence of various types of calls is at the origin of this phenomenon and that, with convenient resource allocation algorithms, this property disappears. In a different context, for circuit switching networks, such a behavior has been identified by Gibbens et al. [8] and Marbukh [15]. In this case, the routing policy is at the origin of metastability. In statistical physics, the metastability property has been known and analyzed for some time. The Ising model, see Liggett [13], is a classical model which has this property with a convenient temperature: for a very long time the spins of most of the particles are up and much later most of them are down and so on. See den Hollander [5] for a survey on these questions. It also illustrates the fact that, as the size of the networks grows, methods of statistical physics should prove to be more and more useful to investigate the complex behavior of these networks. The organization of this paper is as follows: The model for the CDMA cellular network is described in Section 2. In Section 3, a set of experiments is presented. It is shown that, with appropriate parameters, the metastability property holds. Section 4 shows that, in the same situation as in Section 3, a convenient
METASTABILITY OF CDMA CELLULAR SYSTEMS
3
resource reservation algorithm solves the problem. Section 5 presents a simplified mathematical model where it is possible to prove the metastability property. 2. Model description CDMA systems are interference-limited systems. Since the co-channel interference changes according to the loading, this implies that capacity of a cell varies with the traffic load in the cell and neighboring cells. We consider the up-link (transmission of mobiles to base station (BS)) of a CDMA system with multiple cells and services. To achieve the QoS of calls, the energy-per-bit-to-total-interference density (Eb /I0 ) received at a BS for each call should be maintained higher than a predefined threshold depending on its class, in order to satisfied a desired BER. Network state feasibility. We consider a multicell network with N cells and L services classes. A call of class j (1 ≤ j ≤ L) has bit rate transmission Rj , activity factor νj , and requires that a minimum (Eb /I0 ) equal to Γj must be received at a BS to decode the signal of a call from class j. Assuming perfect power control between a BS and mobiles, the energy-per-bit-to-total-interference density at BS i (1 ≤ i ≤ N ) of a mobile with class j is given by Staehle et al. [16] Eb Pj /Rj = I0 i,j N0 + Iiown + Iiother − Pj νj /W where Pj is the received power at BS i of a mobile from class j, N0 is the thermal noise power spectral density, Iiown is the total intra-cell interference density caused by all mobiles inside cell i, Iiother is the total inter-cell interference density due to mobiles in other cells and W is the modulation bandwidth of the system. Then, we have L 1 X Pj νj nij Iiown = W j=1 where nij is the number of mobiles in cell i from class j. For the calculation of the other cell interference at BS i, since the radio environment is not homogeneous in practice, it is assumed that the propagation loss of a mobile in a cell is modeled as the product of the mth power of distance and a log-normal component representing the shadowing losses due to various obstacles (building, trees, cars, etc). If a mobile is at the position x in cell k with BS at location yk then the interference to BS i at location yi is given by Evans and Everitt [6] m kx − yk k 10η/10 Iik (x) = kx − yi k where k · k is the Euclidean norm, m is the path-loss coefficient and η is a centered normal random variable with variance 2σ 2 . Consequently, the average interference caused by a mobile power controlled by BS k in cell i is equal to m Z kx − yk k (ln(10)/10σ)2 (1) αik = e F (dx) kx − yi k Ck where Ck is the region of cell k and F (x) is the location distribution of a mobile in the cell. Thus, Iiother =
1 W
N L X X
Pj νj nkj αik .
k=1,k6=i j=1
Finally, under power constraints of mobiles, where P ∗ is the maximum signal power emitted by a mobile (equal for all classes) [1], a particular network state will be called feasible if (2)
Γj ≤
P ∗ /Rj , N0 + Iiown + Iiother − P ∗ νj /W
1 ≤ i ≤ N, 1 ≤ j ≤ L,
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N. ANTUNES, C. FRICKER, PH. ROBERT, AND D. TIBI
in the sense that radio resources are sufficient to meet the service requirements of all mobiles. When condition (2) is violated for some cell i and class j, we say that such mobiles are in outage. Note also, that the right hand side of (2) is an increasing function of P ∗ . Network State Admissibility. We are interested in the set S of the network states that are feasible. After some manipulations of equations (2), a particular state of the network state (nij , 1 ≤ i ≤ N, 1 ≤ j ≤ L) is feasible (or admissible) if N X L X
(3)
k=1 j=1
where
nkj cikj ≤ min ξj , j:nij >0
1≤i≤N
1 Rj N 0 W νj αik , i 6= k − + νj . = , ξj = νj , i=k R j Γj P∗ The feasible condition in a cell is determined by the active class in cell with the minimum ξj which is a function of the system parameters and the characteristics and QoS constraint of class j. We assume that the decision of admitting or blocking a new or handoff call just depend on the network state at that time. Specifically, such a call is admitted if and only if the resulting network state belongs to S. This may result in excessive QoS degradation of ongoing calls due to the mobility of mobiles in the case of congestion, however as previously mention the goal is not to propose a CAC algorithm but to show the existence of an uncommon phenomena in the network - metastability. Other admission policies can be used, such as, a subset of admission states S to keep a margin in network resources to reduce outage or use a larger set than S where more new calls are admitted even if some mobiles are in outage. This last case may be when one class (e.g. voice) has priority over other class (e.g. video). The proposed model exhibits the main relevant features of a CDMA cellular network for our study. In spite of fast fading being not explicit modeled (its effects could be encapsulated in the (Eb /I0 ) requirements), we claim that the metastability behavior observed in the network remains valid. The next section presents a set of simulations concerning the evolution of the state of the network. It is shown, (Section 5), that this model can be mathematically analyzed to prove the existence of metastability. cikj
3. Metastability phenomena We have considered a WCDMA network with 49 hexagonal cells, as shown in Figure 2(a). BSs have an omnidirectional antenna (1 sector per cell) located at the centers of the hexagonal grid. We assume two types of services: voice (class 1) and video (class 2) calls. New calls of class j ∈ {1, 2} arrive at a cell according to a Poisson process with rate λj . The call duration and cell dwell time of class j calls are exponentially distributed with mean 1/µj and 1/γj , respectively. The destination cell of a handoff call attempt is uniformly distributed between the neighbor cells in both classes. The radio and mobility parameters are given in Table 1. The other parameters of the network are: path-loss coefficient m = 4, shadow fading standard deviation σ = 6 dB and chip rate W = 3.84 Mcps (UMTS). For more details on the choice of parameters, we refer to [9, 10]. By taking cell 25 as a reference in Figure 2(a), the cellular region over which mobiles contribute significantly other to the inter-cell interference in the BS (I25 ), are the mobiles located in the first ring of surrounding cells and the mobiles located in the second ring of surrounding cells. Mobiles lying in cells outside the second ring contribute with a negligible amount to the total interference in cell 25. Approximating the hexagonal cells with circles and supposing that mobiles have uniformly distributed location inside the cell, we have from Equation (1) that α25,i = 0.0474, α25,i = 0.0016 and α25,i = 0.0002, for a cell i in the first, second and outside second ring of cell 25, respectively. Thus, only the interference caused by the two rings that surround one cell is considered. Finally, the simulations results presented used a wrap-around network so that the network has a statistical homogeneity, see Everitt [7]. Figure 2(b) shows the evolution over time of the number of classes 1 and 2 calls in the network. A metastability behavior is identified: the network oscillates between two set of states, stable regions, A and B respectively, with a long residence time in each of them. When the network starts empty, its state moves
METASTABILITY OF CDMA CELLULAR SYSTEMS
Class 1 Bit rate (Rj ) 12.2 kbps Activity factor (νj ) 0.5 Maximum transmission power (Pj∗ ) 32 dBm Minimum energy-per-bit-to-total-interference density (Γj ) 9 dB Call arrival rate (λj ) 1/12 calls/sec Mean call duration (1/µj ) 200 sec Mean cell dwell Time (1/γj ) 400 sec Table 1. Traffic parameters
5
Class 2 144 kbps 1 32 dBm 7 dB 1/27 calls/sec 500 sec 800 sec
rapidly to a set of states where both classes of calls coexist in the network during some period of time Region A. After some time, class 2 calls progressively disappear from the network and the network falls into another set of states where few class 2 calls are present - Region B. When the network is in region B, class 2 calls are very likely rejected by the admission control since they require a significant fraction of the capacity of the cell while class 1 calls are admitted due to their minimal requirements. A rare event, like a very large number of arrivals of class 2 calls or departures of class 1 calls, may result in a large decrease of the number of class 1 calls in some cells and consequently in a significant increase of the number of class 2 calls. In this situation, the state of the network will be in region A. In a similar way, an uncommon event may change the local equilibrium between the number of calls of both classes back to region B. See Figures 3 and 4 for a representation of this bistable behavior. Note that there does not always exists regions A and B, indeed as Figures 5 shows, for some of the values of the parameters, this bistability phenomenon does not occur. Simulations. Several performance measures have been calculated over some time window. From these simulations, it turns out that the metastability phenomenon has a clear impact on the evolution of their estimations. Figures 3(a) and 3(b) present the percentage of new and handoff call blocking rate over periods of length 4 × 103 sec. When the network is in region B, the blocking rate of class 2 calls is almost 1 while the blocking rate of class 1 calls is below the standard values. On the other hand, in region A both classes have a high new call blocking rate (because of the heavy-traffic regime considered here). The blocking rate of class 2 handoff calls has high values in both regions and for class 1 calls in region A. This is mainly due to the simple CAC algorithm which admits new calls as long as network resources are available. Additionally, the high mobility of mobiles (more than 30% the probability of handoff) leads to a high percentages of handoff blocking rate when the network is saturated. Figure 4(a) shows the percentage of admitted calls that are forced to terminate before being completed. The averages are calculated on time intervals of duration 1.5 × 104 sec. The same behavior is shown in the previous figures for new and handoff call blocking is observed for class 1. The values are not so severe in both network states since the majority of calls terminate a call before any handoff. We found larger fluctuations for the percentage of forced termination of class 2 calls when the network is in region B. This is due to the fact that few class 2 calls are admitted while in this region, leading to high fluctuations between the number of calls terminated abruptly by handoff blocking and calls terminated with success. However, forced termination of class 2 calls during the period of time that the network is in region B is clearly higher than in region A. Finally, Figure 4(b) represents the average total bit rate obtained by a class 2 call during its duration (calculated on time intervals of duration 1.5 × 104 sec). The variability when the network is in region B is again due to the very small number of calls admitted during the periods of measure. When the network is in region A, a class 2 call receives in average 68% of its mean total required bit rate. Metastability does not always occur. As mentioned earlier the existence of metastability depends on the parameters of the traffic and of the network. We give an example where a change of parameter affects the global behavior of the network. If we consider that the arrival rate of class 1 calls is now 1/20 calls/sec and that the remaining parameters are the same as in Table 1. Figure 5(a) shows that the percentage of
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N. ANTUNES, C. FRICKER, PH. ROBERT, AND D. TIBI
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(a) Standard cellular layout
500000 1e+06 1.5e+06 2e+06 2.5e+06 3e+06
0
(b) Number of calls in the network versus time
Figure 2. Cellular network and number of calls
1
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(a) Percentage of new call blocking
0 0
500000 1e+06 1.5e+06 2e+06 2.5e+06 3e+06
(b) Percentage of handoff blocking
Figure 3. New and handoff call blocking new call blocking is still high due to the heavy-traffic regime but does not change anymore over time even on a long time scale. The same behavior is observed for the dropping rate of calls (see Figure 5(b)). The poor QoS obtained in forced termination of calls is again a consequence of the simplistic CAC considered.
METASTABILITY OF CDMA CELLULAR SYSTEMS
0.5
7
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220 0
(a) Percentage of dropped calls
500000 1e+06 1.5e+06 2e+06 2.5e+06 3e+06
(b) Transmission rate
Figure 4. Forced termination and average total transmission rate of class 2 calls Because of the oscillations between several set of states with very different performances, the metastability property is an undesirable phenomenon for a network which has to be taken into account in the design of radio resource management schemes. The next section shows that a simple procedure stabilizes the network so that the metastability cannot occur. 4. Resource-reservation scheme In many of existing call admission control algorithms, priority is given to some classes of traffic when the system is congested [14, 4]. In this section we show that if a fraction of network resources (in terms of interference) is reserved for admission of new calls, it can prevent starvation to occur and, consequently, the metastability phenomena may disappear. The basic idea, related to trunk reservation policies used in circuit switching networks (see Gibbens et al. [8]) is to use specific thresholds for the admission of new policy. Choosing a fixed threshold ψj < ξj for class j, would result in admitting less new calls of class j, avoiding to occupies all the available resources and, consequently providing that potential calls from other classes can be admitted in the network. Alternative resource-reservation schemes where thresholds are adjusted adaptively according to radio conditions and traffic conditions in neighboring cells can be found in [14, 4]. The static threshold policy is given as follows: a new call of class j is accepted if the resulting network state (nij , 1 ≤ i ≤ N, 1 ≤ j ≤ L) satisfies (4)
N X L X k=1 l=1
nkl cikl ≤ min ψl , l:nil >0
for all 1 ≤ i ≤ N . The admission of handoff calls remains the same, a call is admitted if the future network state is feasible (belongs to S), given by Condition (3). Since the threshold value for handoff calls is higher than for new calls within a class, the handoff call requests get higher priority over new calls requests. Also blocking more new calls improves the forced termination probability of the calls that are admitted and thus there there is a trade-off between both.
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N. ANTUNES, C. FRICKER, PH. ROBERT, AND D. TIBI
0.85
0.34 Class 2
Class 2 0.32
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0.6 Class 1 0.55
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0.5 0
500000 1e+06 1.5e+06 2e+06 2.5e+06 3e+06
(a) Percentage of new call blocking
0.16 0
500000 1e+06 1.5e+06 2e+06 2.5e+06 3e+06
(b) Percentage of dropped calls
Figure 5. New call blocking and forced termination with λ1 =1/20 calls/sec. Figure 6(a) shows the evolution of the number of calls in the system using the simulation model of Section 3 (see Table 1) with ψ1 = 0.65ξ1 and ψ2 = 0.85ξ2 . The thresholds used to admit new calls prevents that calls of that class 1 gain control of the network due to its high arrival rate and low required bit rate. On the other hand using ψ2 < ξ2 will allow to increase the QoS of ongoing calls. Note that, if ψ1 is much greater than ψ2 then the metastability can persist in the network. These values are dependent on the parameters of the system, otherwise the metastability property may not disappear. Figure 6(b) presents a snapshot of the state of cells where we can see that both classes are presented in the network with an efficient utilization of the system resources. Figure 7(a) and 7(b) show that keeping guard margins in the network resources does not only prevent metastability, but also as expected reduce the percentage of call forced termination with a consequent increase in the average total transmission rate of class 2. The percentage of dropped calls is higher in class 2 since they have a higher mobility pattern, however its total average transmission bit rate over time is almost equal to the required average (500×144 kbps). 5. A mathematical model In this section, we analyze a simple mathematical model of a CDMA cellular network with two classes of calls. It is shown that, under some conditions, it exhibits a bistability behavior. In order to cope with the complexity of the mathematics, it will be assumed that — the network is homogeneous: the traffic at a cell and its characteristics are the same throughout the network; — the network is completely connected. The last condition is quite classical in some related studies in statistical physics. If these assumptions give a very rough mathematical description of the network, a crucial feature of this system is nevertheless preserved: the competition of different classes of calls within each cell of the network and the motion of mobiles within the network. It is shown that in such a context, metastability occurs. These simplifications have also the
METASTABILITY OF CDMA CELLULAR SYSTEMS
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500000 1e+06 1.5e+06 2e+06 2.5e+06 3e+06 (class 1 calls, class 2 calls) (b) State of cells at time of 1.5 × 106 sec
(a) Number of calls versus times
Figure 6. Number of calls and state of cells with resource reservation
0.045 Class 2
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0
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(b) Average total transmission rate in intervals of 1.5×104
sec
Figure 7. Forced termination and average total transmission rate of class 2 calls with resource reservation advantage of showing that the geometry is not at the origin of metastability, despite it may have an influence on its intensity.
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N. ANTUNES, C. FRICKER, PH. ROBERT, AND D. TIBI
It will also be assumed, classically, that the interference from other cells is proportional to the own PN interference, i.e. Iiother = βi Iiown with βi = k=1,k6=i αik (usually βi = 0.5). With this assumption the condition on interferences and Relation (3) give that a state ((ni1 , ni2 ), 1 ≤ i ≤ N ) is admissible when, for any 1 ≤ i ≤ N , the relation ni1 ν1 (1 + βi ) + ni2 ν2 (1 + βi ) ≤ min ξj j:nij >0
holds. In addition it will be assumed that boundary region of admission is linear when ξ1 and ξ2 are similar, C say. Therefore a state ((ni1 , ni2 ), 1 ≤ i ≤ N ) is admissible when (ni1 , ni2 ) ∈ X holds with X = {n1 A1 + n2 A2 ≤ C} . for some constants A1 and A2 . If (n1 , n2 ) ∈ X is the state of a cell of the network, for j = 1, 2, the quantity nj is the number of mobiles of class j at this cell. Statistical Assumptions on the Mobility Model. For class j ∈ {1, 2}, it is assumed that — Call Arrivals: calls of class j arrive at each cell in the network according to a Poisson process with rate λj ; — Call durations: a mobile with a call of class j that is never rejected during its call duration spends an exponentially distributed time with rate µj in the network; — Dwell times: the cell residence of a mobile with service class j in a cell is exponentially distributed with parameter γj ; — Mobility: when a mobile with call of class j leaves a cell, the next cell is chosen uniformly among the other cells. N N (t)) (t), Xi2 The state of the network at time t is described as X N (t) = (XiN (t), 1 ≤ i ≤ N ) where XiN (t) = (Xi1 N N and, for j = 1, 2, the quantity Xij (t) is the number of mobiles of class j in cell i. The process (X (t)) is a Markov process with state space S defined by S = ((ni1 , ni2 )) ∈ N2N : (ni1 , ni2 ) ∈ X , ∀1 ≤ i ≤ N .
The network is analyzed under the assumption that the number of cells goes to infinity. It is shown that the state descriptor (Y N (t)) of the network defined below converges to a continuous dynamical system (y(t)). The stability of the possible equilibrium points of this dynamical system is the key issue. As it will be seen, the existence of more than one stable point gives raise to the metastability property. For n ∈ X , define YnN (t) as the fraction of the number of cells which are in state n at time t YnN (t) = 1{X1N (t)=n} + 1{X2N (t)=n} + · · · + 1{XNN (t)=n} /N in particular Y N (t) = (YnN (t)) belongs to the simplex ( P=
(xn ) ∈ R
X
) :
X
xn = 1 .
n∈X
With these simplifications, the results obtained by Antunes et al. [2] in the context of queueing systems, can be now used in our context of cellular networks to prove the metastability phenomenon. These results are summarized and adapted to our context in the following propositions. Proposition 1 (Convergence to a dynamical system). The sequence (Y N (t)) converges in distribution to the solution (y(t)) of the ordinary differential equation, (5)
y 0 (t) = V (y(t)),
METASTABILITY OF CDMA CELLULAR SYSTEMS
11
where, for y ∈ P, V (y) = (Vn (y), n ∈ X ) is the function Vn (y) =
2 X
λj + γj y (j)
yn−fj 1{nj ≥1} − yn 1{n+fj ∈X }
j=1
+
2 X
(γj + µj ) (nj + 1)yn+fj 1{n+fj ∈X } − nj yn
j=1
with y (j) = m∈X mj ym and f1 = (1, 0) and f2 = (0, 1) are the unit vectors of R2 . A vector y is an equilibrium point of (y(t)) if y = νρ = (νρ (n)) such that P
νρ (n) =
1 ρ1n1 ρ2n2 Z n1 ! n2 !
with Z =
X ρ n 1 ρn 2 1 2 n1 ! n2 !
n∈X
2
and ρ = (ρ1 , ρ2 ) ∈ R satisfies the relation (6) ρj = λj + γj y (j) /(γj + µj ),
j = 1, 2.
Note that if y = νρ , the normalizing constant Z satisfies the relation (7)
ρj
1 ∂Z 1 X ρn1 1 ρ2n2 ∂ log Z = ρj = nj = y (j) . ∂ρj Z ∂ρj Z n1 ! n2 ! n∈X
The next proposition is the key result to identify networks with several stable equilibrium points. Proposition 2 (Stability of Equilibrium Points). The stable points y = (yn ) of the dynamical system defined by Equation (5) are defined by y = νρ and ρ = (ρ1 , ρ2 ) ∈ R2+ are the local minima of the function φ defined by (8)
φ(ρ) = − log Z(ρ) +
2 X
(βk ρk − αk log(ρk ))
k=1
with αk = λk /γk , βk = (γk + µk )/γk . It is very difficult in general to prove the metastability property. In the following, we give a very simple example where it can be proved that the dynamical system has at least three fixed points: Two of them being stable and the other a saddle point. It will be assumed that (A1 , A2 ) defining the state space X are such A1 = 1 and A2 = C so that, at a given cell, there may be n class 1 calls, 0 ≤ n ≤ C, or only one class 2 call. The two classes cannot coexist at a given cell and, moreover, when a cell contains class 1 calls, it has to get completely empty before accommodating a class 2 call. Starting from an initial state where all cells contain only class 1 calls, the network will, very likely, evolve in a stable subspace S1 where the states have few class 2 calls. If the arrival rate λ2 of class 2 calls is sufficiently large then, intuitively, class 2 calls may, in the end, occupy a non-negligible proportion of the cells and therefore reach another stable subspace S2 . Similarly, starting from S2 , due to the pressure of class 1 calls, class 2 calls may progressively disappear from the network to go back to the original situation. This phenomenon does not always happen, there may be only one stable region, depending on the values of the parameters. Proposition 3 (Existence of Metastability). For a network with two classes of calls such that A1 = 1, A2 = C, γ1 = γ2 = 1, µ1 = µ2 = 0, for C sufficiently large, there exist λ1 and λ2 ∈ R+ such that the function φ has at least two local minima. Consequently, the associated network has two stable equilibrium points and hence exhibits metastability. The condition µ1 = µ2 = 0 implies that a call lasts as long as it can, i.e. until it is rejected by a congested cell.
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N. ANTUNES, C. FRICKER, PH. ROBERT, AND D. TIBI
Proof. Fix ρ ∈ R2+ and choose (λ1 , λ2 ) ∈ R2+ so that ρ satisfies Equations (6), i.e. ∂ log Z (k) (9) λk = ρk − νρ = ρk 1 − (ρ) , k = 1, 2, ∂ρk by Relation (7), so that νρ is an equilibrium point for the limiting dynamics. It will be assumed for the moment that C = +∞. The corresponding function φ is then given by ˜ φ(ρ) = − log (ρ2 + eρ1 ) + ρ1 + ρ2 − λ1 log ρ1 − λ2 log ρ2 . Using Equation (9), one gets that ρ2 (ρ2 + (1 − ρ1 )eρ1 ) ∂ 2 φ˜ λ1 ρ2 eρ1 = (ρ) = 2 − 2 ∂ρ1 ρ1 (ρ2 + eρ1 )2 ρ1 (ρ2 + eρ1 )2 and
∂ 2 φ˜ λ2 1 (ρ) = 2 + > 0. ∂ρ22 ρ2 (ρ2 + eρ1 )2 If ρ¯ = (¯ ρ1 , ρ¯1 ) is chosen such that the inequality ρ¯2 < (ρ¯1 − 1) exp(¯ ρ1 ) holds, then ∂ 2 φ˜ ∂ 2 φ˜ (¯ ρ) < 0 and (¯ ρ) > 0. 2 ∂ρ1 ∂ρ22 The constant C is now assumed to be finite and sufficiently large so that the above inequalities with φ in place of φ˜ are satisfied, ρ¯ is a saddle point for φ. The function φ is given by ! C X ρn1 φ(ρ) = − log ρ2 + + ρ1 + ρ2 − λ1 log ρ1 − λ2 log ρ2 . n! n=0 The function ρ2 → φ(¯ ρ1 , ρ2 ) is convex, ρ¯2 is a strict local minimum by construction and therefore a global minimum. Similarly, the function ρ1 → φ(ρ1 , ρ¯2 ) has a strict local maximum at ρ¯1 , inf{φ(ρ) : ρ = (ρ1 , ρ¯2 ), ρ1 < ρ¯1 } < φ(¯ ρ), inf{φ(ρ) : ρ = (ρ1 , ρ¯2 ), ρ¯1 < ρ1 } < φ(¯ ρ) = inf{φ(ρ) : ρ ∈ ∆}, with ∆ = {(ρ¯1 , ρ2 ) : ρ2 ∈ R+ \ {0}}. Since φ((ρ1 , ρ2 )) converges to +∞ when ρ1 or ρ2 converges to 0 or +∞, one concludes that the function φ has at least two local finite minima, one on each side of ∆. The proposition is proved. 6. Conclusion We have shown through experiments that the coexistence of a variety of different traffics in UMTS networks may lead to a metastability property. The main practical implication of this behavior is that QoS is hardly possible to guarantee with a reasonable accuracy in such a context. Moreover, a simplified mathematical representation of the state of the network as a dynamical system with multiple stable points has established the representation of this phenomenon on a rigorous basis. Finally, a CAC that guarantees capacity for the admission of new calls from different classes can prevent metastability. References 1. R. Akl and S. Nguyen, Capacity allocation in multi-cell UMTS networks for different spreading factors with perfect and imperfect power control, Proceedings of the IEEE Consumer Communications and Networking Conference, January 2006, pp. 928–932. 2. Nelson Antunes, Christine Fricker, Philippe Robert, and Danielle Tibi, Stochastic networks with multiple stable points, January 2006, http://arxiv.org/abs/math.PR/0601296. 3. Anton Bovier, Metastability and ageing in stochastic dynamics, Dynamics and Randomness II (Santiago de Chile) (A. Maas, S. Martinez, and J. San Martin, eds.), Kluwer, Dordrecht, 2004, pp. 17–81. 4. H. Chen, S. Kumar, and C.-C. J. Kuo, QoS-aware radio resource management scheme for CDMA cellular networks based on dynamic interference guard margin, Computer Networks 46 (2004), 867–879. 5. F. den Hollander, Metastability under stochastic dynamics, Stochastic Processes and their Applications 114 (2004), no. 1, 1–26.
METASTABILITY OF CDMA CELLULAR SYSTEMS
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6. J. Evans and D. Everitt, On the teletraffic capacity of CDMA cellular networks, IEEE Transactions on Vehicular Technology 48 (1999), no. 1, 153–165. 7. D. Everitt, Analytic traffic models of CDMA cellular networks, Proceedings of the 14th International Teletraffic Congress (France), June 1994, pp. 349–356. 8. R. J. Gibbens, P. J. Hunt, and F. P. Kelly, Bistability in communication networks, Disorder in physical systems, Oxford Sci. Publ., Oxford Univ. Press, New York, 1990, pp. 113–127. 9. H. Holma and A. Toskala, WCDMA for UMTS: Radio access for third generation mobile communications, John Wiley & Sons, 2002. 10. W.S. Jeon and D.G. Jeong, Call admission control for CDMA mobile communications systems supporting multimedia services, IEEE Transactions on Wireless Communications 1 (2002), no. 4, 649–659. 11. I. Kim, B. Shin, and D. Lee, SIR-based call admission control by intercell interference prediction for DS-CDMA cellular systems, IEEE Communications Letters 4 (2000), no. 1, 29–31. 12. W.C.Y. Lee, Overview of cellular CDMA, IEEE Transactions on Vehicular Technology 40 (1991), 291–302. 13. T.M. Liggett, Interacting particle systems, Grundlehren der mathematischen Wissenschaften, Springer Verlag, New York, 1985. 14. D. Liu, Y. Zhang, and S. Hu, Call admission policies based on calculated power control setpoints in SIR-based powercontrolled DS-CDMA cellular networks, Wireless Networks 10 (2004), 473–483. 15. Vladimir Marbukh, Loss circuit switched communication network: performance analysis and dynamic routing, Queueing Systems. Theory and Applications 13 (1993), no. 1-3, 111–141. 16. D. Staehle, K. Leibnitz, K. Heck, B. Schrder, A. Weller, and P. Tran-Gia, Approximating the othercell interference distribution in inhomogeneous UMTS networks, Proceedings of the IEEE Vehicular Technology Conference (Birmingham, AL), May 2002, pp. 1640–1644. (N. Antunes) Universidade do Algarve, Faculdade de Cincias e Tecnologia, Campus de Gambelas, 8005-139 Faro, Portugal E-mail address:
[email protected] (C. Fricker, Ph. Robert) INRIA, domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France E-mail address, C. Fricker:
[email protected] E-mail address, Ph. Robert:
[email protected] URL, Ph. Robert: http://www-rocq.inria.fr/~robert (D. Tibi) Universit Paris 7, UMR 7599, 2 Place Jussieu, 75251 Paris Cedex 05, France E-mail address:
[email protected]
