CAPACITY OF A CDMA SYSTEM ENSC 833 NETWORK PROTOCOLS AND PERFORMANCE PROF. STEPHEN HARDY SIMON FRASER UNIVERSITY

Transcription

1 CAPACITY OF A CDMA SYSTEM ENSC 833 NETWORK PROTOCOLS AND PERFORMANCE PROF. STEPHEN HARDY SIMON FRASER UNIVERSITY Brad Zarikoff zarikoff@ieee.org December 1, 2005

2 Table of Contents Introduction...3 I. Literature Review...3 A. CDMA Outage Probability...3 B. Erlang Capacity Single Service CDMA Multi-service CDMA...5 C. CDMA Call Blocking Probability Single-service CDMA Multi-service CDMA Multi-service Uplink and Downlink CDMA Multi-service with Loading Factor...7 II. Identification of Open Research Problems...7 III. CDMA Access Model...8 IV. Method to Calculate Effective Capacity...9 V. Results...14 A. Effective Capacity Simulations...14 B. Erlang Capacity Formula...17 VI. Conclusions...19 VII. References...19 Table of Figures Figure 1: Two-dimensional Markov chain...10 Figure 2: State transitions from state (n,k)...10 Figure 3: All users are active (n=k)...12 Figure 4: No users are active (k=0)...12 Figure 5: General state, n users accepted, k users active...12 Figure 6: Illustrated recursive method to find P nk...13 Figure 7: Number of users, variable arrival rate λ...15 Figure 8: Number of users, variable activity factor ε...16 Figure 9: Number of users, variable number of interferers N s...17 Figure 10: Activity factor versus capacity

3 Introduction For the ENSC 833 class project, I have chosen topic 1: the analysis of capacity, performance, and planning of multimedia CDMA cellular networks. In particular, I will focus on CDMA system capacity. The original cellular technologies were based solely on time and frequency multiplexing. They were easy to implement, with performance based on the magnitude of the channel noise. However, they were also limited in that each user would take a fixed and equal amount of spectrum during a call. Once all frequency or time slots were full, all subsequent users were blocked; the Erlang formulae apply equally here as they did for the circuit switched land-line telephone system (POTS). The technique of code division multiple access (CDMA), or spread spectrum, created a new type of channel. Each link in a CDMA channel is limited by interference between concurrent users instead of noise. This allows for a system to have a soft-limit on the number of users; as the number of users increases in a CDMA system, the overall system interference increases. Given a set performance level, a CDMA system can provide for a fixed number of users. However, the overall performance could be lowered at the expense of an increased number of users. Most CDMA systems operate at a maximum capacity dictated by a minimum outage probability. The maximum number of users can be calculated by taking into account the number of neighbouring cells, the number of users per cell, the receiver noise, the channel bandwidth and the transmit power. This capacity was initially based on continuous usage with fixed interference levels, where each user was assumed to be actively transmitting at all times. The inter-cell interference levels were modeled by a factor f. The blockage probability, Erlang capacity and outage probability is of more interest to a system designer; they are based on a statistical model of the users and interference. I. Literature Review A. CDMA Outage Probability Since CDMA system performance is dependent on interference levels, the number of users active in the system at any time is the limiting factor on capacity. 3

4 The relationship between the number of users and the outage probability is necessary to find the maximum number of users that can be accommodated in a CDMA system. By fixing the outage probability, the system can guarantee a quality of service. In [1], Gilhousen applies statistical models to the traditionally static outage probability expression. Specifically, [1] assumes that the inter-cell interference is a Gaussian random variable, and that the user activity within a cell (whether or not a user is actively accessing the channel) is modeled as a binary random variable. The expression derived in [1] for outage probability is: W R η0 k E[ I S] Eb N S 0 Pout ( k) = Q var[ IS] where W R = processing gain (channel bandwidth/user rate) Eb N0 = necessary user SNR for BER performance of 10 S η0 = received SNR per user k = number of active users in the current cell IS= inter-cell interference ratio As mentioned above, the inter-cell interference ratio is assumed Gaussian, with the following values (from [1]): where [ ] [ ] EIS N var IS N, N s is the number of users per cell. s The final expression for outage probability in [1] was marginalized over each user s activity random variable. B. Erlang Capacity 1. Single Service CDMA The Erlang capacity in a CDMA cell was first analyzed for the reverse link in [2]. Viterbi modified the traditional CDMA expression that related user signal to interference and noise ratio (SINR) to the number of users, bandwidth and the inter-cell interference. Viterbi s model consists of two random variables: a voice s 3 4

5 activity factor, ε, which was modeled as a binary random variable (same as in [1]), and the received SINR for each user in the current cell and neighbour cell, which was modeled as a log-normal random variable (compared to a Gaussian random variable in [1]). This log-normal approximation was justified using experimental results. The effect of the two different models on the cell capacity is studied in [3], where it is shown that the Gaussian approximation produces the higher maximum capacity. The number of active users was assumed to be Poisson distributed, with a mean of ρ = λµ ; this is also the Erlang capacity. After simplification, the resulting Erlang capacity formula was found to be: λ = µ ε where ( 1 η )( ) ( 1+ f)( E I ) W R F b 0 median Erlangs/Sector η = nominal noise to interference ratio W = spread-spectrum bandwidth R = data rate F = reduction factor ε = probability of user being 'on' f = inter-cell interference factor E I 0 = SINR per user ( ) b median F is dependent on the probability of blocking (Section I.C.1.), the noise to interference ratio η (taken to be -10dB in [2]), and the standard deviation of lognormal distributed Eb I 0. Notice that Viterbi has approximated the inter-cell interference by the factor f, but has kept the log-normal effect of the current cell users in F. The total system utilization, in Erlangs per sector, can thus be expressed as a function of the CDMA system parameters (W, R, f) and the statistics of the SINR per user. Notice that there is a linear relationship between the activity factor (the probability that a user is in an active state) and the Erlang capacity. For some numerical examples, see [4]. 2. Multi-service CDMA [5] examined the case of a system with multiple services (i.e. a CDMA system handling both voice and data, where both have independent service arrival and 5

6 departure rates). The method involved first determining the peak number of calls that the system could accommodate using traditional CDMA capacity formulae, where the number of active call s were assumed static and the activity factor was set to a constant 1. This peak number of calls (or trunks) was then partitioned and a sharing model designed to accommodate the two services. C. CDMA Call Blocking Probability 1. Single-service CDMA As part of calculating the Erlang capacity of a CDMA cell, [2] calculated the probability of blocking for the cell. Viterbi calculated both a modified Chernoff upper bound and a Gaussian approximation to the call blocking probability of a single cell, taking into account the user interference within the current cell and disregarding the inter-cell interference. The Gaussian approximation involved applying the central limit theorem to the number of interferers, specifically to the sum of the product of each users SNR and activity factor random variables. 2. Multi-service CDMA Later, in [6], a multi-service CDMA network access protocol was analyzed. New constraints included the inclusion of handoff call arrivals and departures (although there is no provision for call continuity, i.e. handoffs are treated like new calls), and multiple types of services with different service arrival rates, departure rates, and capacity requirements. However, the CDMA channel itself was assumed to have static capacity, with a fixed number of CDMA channels. Three different service sharing protocols were investigated. They were defined as complete sharing, complete partitioning, and partial sharing. The blocking probabilities were calculated in each case. The number of services in the example was limited to two. In complete sharing, there are no reserved states for the incoming calls. Complete sharing results in direct competition for resources for each service; as [6] demonstrates, the result is an unfair allocation of resources if one of the services were to have a higher arrival rate. In complete partitioning, a maximum number of calls can be accepted for each service, effectively separating the services into independent queues. This results in a fairer allocation of resources, but with a loss in total capacity. In partial sharing, while some of the capacity is reserved for each specific service, a portion of the remaining capacity is allocated for use by either service. This allowed for greater 6

7 fine-tuning of capacity allocation. Techniques involving call preemption and discouraged arrivals with the partial sharing technique were also analyzed. 3. Multi-service Uplink and Downlink CDMA To account for both uplink and downlink calls in the same frequency band, [7] analyzed a time-division CDMA network where each CDMA channel was divided into a number of TDMA slots. The analysis revolved around UMTS services. The blocking probability for both circuit calls (real-time) or packet calls (non-realtime) were calculated. Each type of service required a specific load for the uplink and downlink, and each has a separate arrival and departure rate. The optimum allocation of the CDMA uplink and downlink channels was derived through simulation. 4. Multi-service with Loading Factor A recent paper by Navaie [8] calculates the blocking probability in multi-service CDMA by analyzing a loading factor for each service. The loading factor for user i, η i, is dependent on the users SINR, activity factor, and data rate. Specifically, the probability of a blocked call for service j is: ( ) = > ( ) PB j Pr ηul ηth η j where η UL is the loading factor of all users and all services in all cells, η th is the maximum uplink load and η ( j ) is the loading factor for the j th service. Inter-cell interference was modeled in η UL as a factor f. II. Identification of Open Research Problems After the literature review, potential research problems were identified. They include: 1. calculating the Erlang capacity of a CDMA cell given a specific access model, and compare to the single-service case 2. investigate possible methods of maximizing capacity through access model design in single and multi-service systems 7

8 3. application of the statistical model of CDMA system capacity in [2] to the multi-service model with fixed channel assignment in [6] (make the fixed channel assignments statistical) 4. application of the CDMA channel statistics to estimate the short-term maximum occupancy of the cell Problems 1 and 2 are both linked in that to maximize the capacity of a CDMA cell, the Erlang capacity must be calculated. Problem 3 consists of applying what was learned in [2] to the multi-service problem on [6]. Problem 4 is interesting; given the statistical nature of CDMA channels, the maximum occupancy may be considered a random variable in itself, depending on the services and users accessed the channel. A method to predict future capacity based on current levels of inter-cell interference and user activity seems feasible. In fact, [9] handles the single-service case, where the service is either voice or video. Randhawa calculates the residual capacity one slot in the future, where the residual capacity is the number of calls that can be accepted without violating the quality of service. Making some assumptions about future service needs, this may be extended to multiple time slots. This is interesting since it may allow for better resource allocation. For the remainder of this project, problem 1 will be the focus, with a goal of analyzing the maximum number of simultaneous users (and thus the Erlang capacity) in a CDMA channel. The access model described in [9] will be analyzed. The CDMA link capacity formula from [2] will be used for comparison. III. CDMA Access Model When examining a CDMA queuing model, it is important to note the relationship between the capacity and the blocking probability, P B. Blocking in CDMA only occurs if the number of users in the cell has already reached maximum capacity. The capacity itself is determined using outage probabilities and activity factors. So, to determine P B, it is first necessary to determine an expression for the capacity (such as in [1]). The CDMA voice network model in [9] consists of a number of voice channels. A specific activity period ε is assumed, as in [2] and [1]. The higher the activity factor for each user, the lower the overall capacity, since the number of 8

9 simultaneous users will be higher. In [9] however, the activity factor is based on exponentially distributed random variables instead of binomial. From [9], the number of accepted calls is defined as n and the number of nk, denotes accepted calls which are in an active state as k. The Markov state ( ) the state where n calls are established and k of them are currently transmitting information (and therefore are interfering with each other). The call arrivals and departures between these states are represented as a two-dimensional Markov chain, with a Poisson distributed call arrival rate with mean λ and exponentially distributed departure rate with mean 1 µ. The exponentially distributed activity factor has an average activity duration 1 α. Conversely, the exponentially distributed inactivity duration has an average of 1 β. The variables are summarized here: n = number of accepted calls (users currently assigned a channel) k = number of calls in an active state λ = the call arrival rate µ = the call departure rate α = the call activity rate β = the call inactivity rate t = slot length IV. Method to Calculate Effective Capacity Given that the k n, the Markov chain of [9] can be represented as: 9

10 Figure 1: Two-dimensional Markov chain State transitions are discretized and occur only at the beginning of each slot ( t is arbitrarily set to 1 second). The transition rates are easily calculated given nk, as the Markov chain in [9]. The possible transitions are shown for the state ( ) follows: Figure 2: State transitions from state (n,k) 10

11 This model allows us to analyze the CDMA channel access using product form techniques [10]. In [9], Randhawa uses the access protocol described in Section III to determine the residual capacity. The residual capacity consists of the total number of excess users that the system can accept given an outage probability of 1%. The outage probability is computed to be the probability that the bit error 3 rate (BER) exceeds a level of 10, as described in Section I.A. The resulting equations provide a means of predicting the residual capacity using the system parameters and the current system state. To begin the search for the maximum number of users given the outage probability, the state probabilities first need to be calculated. They are defined as P = Pr n = i, k = j. They are used both for normalization and for calculation of ij ( ) the outage probability. The limiting states (where there are either no active users, or there are only active users) will be used for the derivation of the product form solution [10]. The balance equations are found as follows: 11

12 Figure 3: All users are active (n=k) λα Pnk = Pn + 1, k + 1 µ k + α + β P n+ 1, k+ 1 nk P () 1 nk nk n 1, k 1 ( 1) λα = P µ α β = P or = P k ( k + 1)( + ) ρα ( k + 1)( α + β) ρα ( α + β) λβ Pn,0. µ n = Pn 1,0 α + β P λβ = P µ n α β n,0 n 1,0 = Pn 1,0 n ( 2) ( + ) ρβ ( α + β) Figure 4: No users are active (k=0) nk ( µ ( ) µ ) P n k + k λβ λα = Pn 1, k + Pn 1, k 1 α + β α + β ρ = + α + β ( Pn 1, k β Pn 1, k 1 α ) ( 3) Figure 5: General state, n users accepted, k users active 12

13 These equations for P nk will be used to calculate the state probabilities in a recursive fashion. It should be noted that a steady state form of P nk may be achievable; however, as of the due date of this project, it was not found. The recursion can be executed in the following manner. The two sets of edge states, P nn from (1) and P n,0 from (2), can be calculated using an arbitrary P 00. The interior states can then be filled in using the expression for P nk from (3). The graphic below illustrates the recursive process. Figure 6: Illustrated recursive method to find The states enclosed in circles are calculated with (1), in squares with (2) and in stars with (3). Once all the states are calculated for a given N, the normalization can be done, with P 00 N n = 1 P. n= 0k= 0 nk To calculate the effective capacity, the method in [9] is modified. The excess capacity set to 0. Using the expression for outage probability in Section I.A., the relationship between the number of active users and the outage probability is: P nk 3 ( BER > ) = ( P ( k n k ) Pout ( k )) + ( P( k 1 n, k) Pout ( k 1) ) ( P( k 1 n, k) Pout ( k 1) ) Pr 10, Defining the maximum number of accepted users as N, the method is as follows: 13

14 n = 0 1. n = n Calculate the P 's nk 3 ( BER > ) 3 ( BER > ) 3. Calculate Pr If Pr , goto 1. else N = n. V. Results The simulations were done using similar values to those in [9] and [1]: W = 1.25MHz = B 8 kbps Eb = 5 = 7 db necessary for = 10 N 0 Eb 10 N0 3 ( BER ) η = 10 = 0.2 η0 = = 1dB S Ns = 10 α α β ε α β λ = µ = ( + ) = = 0.4 ( activity factor, = 1 20, = 1 30) The number of users in a neighbouring cell was set to Ns = 10. With Eb N 0 set to 7dB, the value for nominal interference to noise ratio in the Erlang CDMA capacity formula was changed to 0.2. The activity probabilities, α and β, were set to insure that the probability of voice activity was ε = 0.4. The outage probability was set to 1%. A. Effective Capacity Simulations Curves for a variety of arrival rates, voice activity factors, and number of users in neighbouring cells were generated to see the effect on the user capacity. First, the dependence of the maximum number of users on the arrival rate was investigated. Since the arrivals do not effect the actual utilization of the CDMA 14

15 channel, it is expected that the arrival rate will have little or no effect on the capacity. The utilization was varied from to 178.2, with the nominal utilization from above being ρ = λµ = The curves below show that the utilization has a limited effect on the maximum occupancy. The capacity of the system stayed relatively constant at N = 50. Since this is equivalent to having 50 simultaneous calls in the system, the capacity can be said to equal to 50 Erlangs. Figure 7: Number of users, variable arrival rate λ Next, the activity rate was allowed to vary, from α = β 4 to α = 4β. The nominal activity factor from above is α = 2β 3. As mentioned in Section I.B.1. with regards to the Erlang capacity formula for single-service CDMA, it is expected that with a higher activity factor, the capacity will drop. This is because the interference level reaches a maximum if each call is continuously active. As expected, the change in the activity factor resulted in a large change in the capacity. The left-most curve below corresponds to ε = 0.8, and results in a capacity of 30 users. With ε = 0.2, the capacity is increased to 92 users. 15

16 However, this results in an impractical activity factor of only 20%. With the nominal voice activity factor of 40%, the capacity is again 50 users. Figure 8: Number of users, variable activity factor ε As a final experiment, the number of users in neighbouring cells were varied, with N = 1, 5,10, 50. This increase has the effect of increasing the outage s probability. 16

17 Figure 9: Number of users, variable number of interferers B. Erlang Capacity Formula If we use the expression for Erlang capacity from Section I.B.1., we expect the outcome to be very close to the results generated through simulation; the difference in the models lie in the modeling of the inter-cell interference and the voice activity factor. Values for the Erlang capacity formula are as follows: η = 0.2 W R = ε = 0.4 activity factor f = 0.43 = 2.47 ( ) Eb I 0 median ( ) The values are based on the same values used for the CDMA access protocol in Section IV.A. W R, Eb I 0, and ε are directly from the CDMA queue. η is simply 1Eb N 0 = 15, and shows the minimum allowable SNR per user to achieve a given performance (in our case, 7dB, as previously stated). Note that the N used to calculate the mean interference is set to 10. The interference s factor f is dependent on an approximation of the inter-cell interference, as noted N s 17

18 in Table 1 of [2]. Finally, the reduction factor F is based on the statistics of the current cell interference. The resulting capacity for ε = 0.4 is ~56.25 Erlangs. This is only slightly higher than the queue capacity found above, which was 50 Erlangs. It is instructive to note the dependence of the capacity formula on the utilization and the activity factor. Similar to the queue protocol capacity results, the CDMA capacity formula is not dependent on the utilization and is inversely dependent on the activity factor. To show the relationship of both techniques on the activity factor, the Erlang capacity for both is plotted below. The Erlang capacity of the CDMA access protocol is again defined as the number of accepted users, since every user in the system is a separate call. Figure 10: Activity factor versus capacity In both cases, the effect of an increase in activity factor is obvious: increase the activity factor, and the overall capacity drops since the users in the system must compete with greater interference, dropping the achievable throughput. 18

19 The access protocol from [9] has a slightly lower capacity, but still follows the Erlang capacity formula. It was reported in [3] that a model with a Gaussian assumption for the inter-cell interference resulted in higher capacities than a model using a log-normal assumption. However, we used the Gaussian assumption in our calculated capacity (in the form of the outage probability from [1]) and found that the log- Normal assumption (in the form of the Erlang capacity in [2]) resulted in higher capacities. The reason for this discrepancy is unclear at the time or writing this paper; however, it is assumed that the queue model is negatively influencing the mean capacity. VI. Conclusions A literature review of the capacity of single and multi-service CDMA systems was completed. A number of open research problems were listed. The problem of calculating the Erlang capacity of the access protocol in [9] was chosen. The single-service CDMA capacity formula from [2] was used as a benchmark for comparison. It was found that the capacity simulations based on the access protocol follows the trend of the Erlang capacity formula, with a slight loss in capacity. This is contrary to the results shown in [3]. The discrepancy is attributed to the queue model. Further study should include an analysis of this problem. The results may lead to a method to improve the queue in [9] and increase its capacity. VII. References [1] K. S. Gilhousen, I. M. Jacobs, R. Padovani, A. J. Viterbi, L. A. Weaver, Jr., and C. E. Wheatley, III, On the Capacity of a Cellular CDMA System, IEEE Trans. Vehic. Technol., vol. 40, pp , May [2] A. M. Viterbi, and A. J. Viterbi, Erlang Capacity of a Power Controlled CDMA System, IEEE J. Select. Areas Commun., vol. 11, pp , Aug [3] C. Kumpairee, C. Kumpan, O. Pin-Ngern, N. Hemmakorn, and S. Noppanakeepong, Blocking Probability of the Reverse Link Cellular CDMA System, IEEE Asia-Pacific Conf. on Circuits and Systems, vol. 2, pp , Oct [4] J. S. Lee, and L. E. Miller, On the Erlang Capacity of CDMA Cellular Systems, IEEE Global Telecommun. Conf., vol. 3, pp , Nov

20 [5] W. Matragi, and S. Nanda, Capacity Analysis of an Integrated Voice and Data CDMA System, IEEE Veh. Tech. Conf., vol. 2, pp , May [6] T. S. Randhawa, and R. H. S. Hardy, Performance Analysis of Multi-Service Cellular Networks with Mobile Users, IEEE Wireless Commun. And Networking Conf., vol. 3, pp , Sep [7] H.-S. Cho, J. Shin, and Y. Lee, Call Blocking Probability for Heterogeneous and Asymmetrical Traffics in a TD-CDMA System, IEEE Commun. Letters, vol. 8, pp , Dec [8] K.Navaie, and F. Fadaie, A Novel Approach to Call Blocking Probability Evaluation in Multiservice CDMA, Canadian Conf. on Elec. And Comp. Engr., vol. 1, pp , May, [9] T. S. Randhawa, and S. Hardy, Traffic Prediction Based Access Control in Wireless CDMA Networks Supporting Integrated Services, IEEE Wireless Commun. And Networking Conf., vol.3, pp , Mar [10] J. F. Hayes, and T. V. J. Ganesh Babu, Modeling and Analysis of Telecommunications Networks, John Wiley & Sons: New Jersey,