Analysis of the Tradeoff between Delay and Source Rate in Multiuser Wireless Systems
© Beatriz Soret et al. 2010
Received: 25 January 2010
Accepted: 3 August 2010
Published: 12 August 2010
This work addresses the limits on the information that can be transmitted over the wireless channel under the conditions stated by the MAC layer: a selected scheduling discipline and an ensured level of QoS. Based on the effective bandwidth theory, the joint influence of the channel fading, the data outsourcing process, and the scheduling discipline in the QoS metrics are studied. We obtain a closed-form expression of the vector of attainable users' rates for several scheduling algorithms, representing the maximum constant rate that the uth user can transmit under the selected discipline and fulfilling a target Bit Error Rate (BER) and the delay constraint given by the pair , where is the target delay and is the probability of exceeding .
Providing Quality of Service (QoS) guarantees to different applications is an important issue in the design of next generation of high-speed networks. The QoS metrics of interest are likely to vary from one application to another, but are predicted to include measures such as throughput, Bit Error Rate (BER), and delay. Unlike traditional data communication, where system performance is largely measured in terms of the average overall throughput and loss rate, real-time communications may require QoS metrics expressed in terms of the mean delay or its variance (jitter).
Traditional networking approaches design separately, the physical and the medium access layer (MAC). Instead, in future wireless networks the physical knowledge of the wireless medium is shared with higher layers on a cross-layer basis , an increasingly important topic for the evolving wireless build-out.
User multiplexing for QoS guarantees is an active research topic  in wireless systems, under different names such as, subcarrier and slot allocation, resource allocation or scheduling. Exploiting both the source diversity and the variations in channel conditions can increase the system throughput. A scheduling scheme ideally should be able not only to handle the uncertainty of the channel but also to exploit it, that is, opportunistically serve users with good channels. Using such an approach leads to a system capacity that increases with the number of users (multiuser diversity) .
Many questions regarding the performance of most used opportunistic algorithms are still open. For example, very few works consider the delay or study the treatment given to each user [4, 5]. The main difficulty in obtaining analytical results comes from the fact that the classical queueing theory is no longer suitable. Moreover, the result is linked to the scheduling discipline and the analysis has to be done algorithm by algorithm. To the best of our knowledge, the papers with analytical results found in the literature either use simple channel models [6, 7] or only provide bounds on the QoS metrics [8, 9].
Within this context, we explore the limits on the information that can be transmitted over the wireless channel under the conditions stated by the MAC layer: an ensured level of QoS and a selected scheduling discipline. In particular, both the channel fading and the scheduling algorithm determine the maximum source rate to be transmitted under some statistical QoS guarantees.
In single user systems, ergodic capacity  is not a suitable information-theoretic measure for delay sensitive applications over fading channels. On the other hand, delay-limited capacity becomes zero for Rayleigh channels. In this case, its probabilistic version, the Capacity with Probabilistic Delay Constraint, becomes useful . Then, rather than ensuring a deterministic delay, a probabilistic delay constraint is defined as the pair , where is the target delay and is the probability of exceeding it.
On the other hand, in multiuser communications the capacity of the channel is no longer fully characterized by a single number. Instead, the capacity should be redefined to consider each user's data rate separately. Thus, in a system with users, the capacity should take the form of a dimensional vector representing rates allocated to the users . A capacity region is then defined as the set of all dimensional rate vectors that are achievable in the channel. Furthermore, when a QoS constraint is imposed (in the form of a probabilistic delay constraint), the data rate attainable by each user will be closely linked to the scheduling strategy.
In this paper, we obtain a closed-form expression of the vector of users' data rates for several scheduling algorithms, representing the maximum constant rate that the th user can transmit under the selected discipline and fulfilling a target BER and the delay constraint given by the pair . The total system capacity will be the sum of the individual users' rates, where each user can have a different delay constraint and can experience a different channel. The procedure to obtain these rates, based on the effective bandwidth theory , is similar to some previous results in a single-user system . Three simple and widely employed disciplines have been analyzed: Round Robin , Best Channel , and Proportional Fair . For simplicity, the results are obtained for a CBR (Constant Bit Rate) data source but they can be extended to any other traffic model as explained in .
The remainder of the paper is organized as follows. Section 2 describes the multiuser system model. Section 3 first details the derivation of the maximum users' rates subject to a delay constraint for an uncorrelated Rayleigh channel (Section 3.1). Later on, the expressions are particularized to Round Robin, Best Channel, and Proportional Fair disciplines in Sections 3.2, 3.3, and 3.4, respectively. The time-correlated channel is examined in Section 4, first of all the achievable rates (Section 4.1) and then the same three particularizations for the three examples of discipline (Sections 4.2, 4.3, and 4.4). The validation of the results by comparison with simulations is presented in Section 5. Finally, concluding remarks are given in Section 6.
2. Multiuser System Model
2.1. Queueing Model
Physical time is divided into units, hereinafter referred to as symbol periods, which represent the transmission discrete time unit, . The channel response of each user is assumed to be constant over the symbol. Moreover, the scheduler allocates the channel to users in a symbol per symbol basis: every new symbol, a user is selected for transmission.
It is assumed that the transmitter employs adaptive techniques, so that the transmission rate is modified dynamically, seeking to adapt to the time-varying conditions of the physical channel.
Notice that in the sum above only one of the terms is nonzero, corresponding to the user allocated to the channel.
2.2. Channel Model
Every user experiences a flat Rayleigh channel with complex channel response . The envelope of the channel response is denoted by . Furthermore, users are independent among them, that is, the channel response seen by one user is independent from the rest.
where , under adaptive modulation, is a constant related to the target BER. Its value for uncoded QAM is  , where is the target BER. Further, the value represents the upper bound corresponding to the evaluation of the AWGN channel capacity (in Shannon's sense).
The use of the exponential model simplifies and speeds up the simulations and numerical evaluations along this paper without altering the conclusions. Nevertheless, any other correlation function can be used (i.e., the classical Jakes' model ).
2.3. Effective Bandwidth Analysis
Since and are independent of each other, may be decomposed into two terms, , where and are the log-moment generating functions of the accumulated source process and the accumulated channel process of user , respectively.
The procedure to generalize the results to other traffic sources can be found in .
3. Uncorrelated Channel
We start the analysis of the multiuser system presented above with the case of users experiencing an uncorrelated Rayleigh channel (block fading model). Part of these results can be found in .
3.1. Achievable Users' Rates with a Delay Constraint
Starting from (16), the set of individual users' rates that accomplish a delay constraint can be calculated. Each user has his own delay constraint. The effective bandwidth of the channel is needed, .
3.2. Round Robin
It is known that this strategy does not work well over varying channels and a low efficiency in terms of system capacity and QoS differentiation is expected.
The expressions above make it possible to evaluate the individual users' rates in (20) under a Round Robin discipline.
First of all, let us observe the common behaviour of R for all the user. As presumed, the curve increases with (the more relaxed the QoS requirements, the higher the maximum attainable rate). Obviously, R is always below . Moreover, values of the user rate equal to zero must be interpreted as QoS requirements that cannot be fulfilled with that channel conditions, number of users and discipline.
Observing the differences among users, those with better channel conditions obtain higher rates and can demand stringent QoS conditions, as it was expected. Thus, the best user in this example could fix a delay constraint with a target delay of symbols in contrast to the symbols of the worst user. On the other hand, if the same target delay is fixed for the three users the rate to be employed increases for better users (users with better channel conditions). For example, for symbols user can transmit bits/symbol, user , bits/symbol, and user , bits/symbol.
3.3. Best Channel
This algorithm maximizes the total system efficiency. However, under this strategy, good average SNR users get more average throughput than low SNR users.
The differences among users are much more noticeable than for RR. Thus, the best user is better off with the change to BC allocation at the expenses of users with lower average SNR. Notice that not only the differences in the mean are remarkable (the asymptotic behaviour when relaxing the QoS constraint) but also the minimum target delays of each user move away. For example, the worst user cannot demand a target delay below symbols for these channel conditions and scheduling, in contrast to the symbols of the best user. As expected, the average rate is higher than for RR, since this algorithm maximizes the total system efficiency.
3.4. Proportional Fair
It can be observed that the differences among users reduce if we compare with the BC strategy. That is exactly the goal of this discipline: to maintain a balance between the total throughput and the level of service of all users. Obviously, the average rate reduces to increase the fairness. The achieved fairness is specially noticeable in the behaviour of the target delay, which is symbols for the three users.
4. Correlated Channel
In this section, a time-correlated channel is considered, meaning that the channel response of each user follows the exponential ACF described in (8).
4.1. Achievable Users' Rates with a Delay Constraint
The channel correlation among the elements in the block is considered but, with the proper selection of the block's length, large enough, independence among blocks may be assumed. The choice of will be closely related to the correlation of the channel. If the channel is strongly correlated, longer blocks have to be defined in order to assume independent blocks. Whatever the value of is, there is a residual value of correlation between the last elements of one block and the first elements of next one. Nevertheless, this border correlation is negligible when the value of is large enough. Notice that a decreasing autocorrelation function is required, as it is the case in fading channels.
Under these conditions (sufficiently long and ), is the sum of a sufficiently large number of independent random variables, and the Central Limit Theorem can be applied. Thus, approximates a Gaussian random variable with average and variance , where and are the mean and the variance of a block of size of the th user.
On the other hand, the Lilliefors test  is used in statistics to test whether an observed sample distribution is consistent with normality. In the numerical results and simulations conducted throughout this paper, the validity of the Gaussian approximation for has been validated by testing for normality with the Lilliefors test for the selected values of and .
Let us examine first the single user system, since this result will be needed later. With only one user ( ) and continuous rate policy, the mean and variance of the blocks, denoted as and , are calculated as follows.
4.2. Round Robin
4.3. Best Channel
Similarly as done in RR, the calculation of the maximum attainable rates in the case of Best Channel strategy leads to the computation of the variance of the blocks (the evaluation of the mean of the blocks is straightforward), which comes down to the computation of the expectation .
This random variable, equivalent to the effective SNR defined in the uncorrelated channel, indicates the fact that the user only gets the channel if his instantaneous SNR is the highest among all the users. In contrast to the uncorrelated channel, we include the time through the subindex since it is needed to calculate the expectation evaluated in two different symbols.
4.4. Proportional Fair
5. Simulation Comparison
The analytical results presented in Sections 3 and 4 are validated by comparison with simulations. In particular, the queueing system in Figure 1 is simulated. Each user sends bits to its buffer of queue length in the th symbol. The selected user depends on the scheduling algorithm: Round Robin, Best Channel, or Proportional Fair. , and are fixed (the same for all the users, for simplicity) and the users' rates are evaluated with (20). Then, the the arrival process of each user generates source data at the (constant) rate R . The simulation is run and the tail probability of exceeding the target delay is measured based on the measurements of the delay suffered by bits leaving the queue. Notice that the expected value of this tail probability, , is .
5.1. Uncorrelated Channel
The results show that the QoS requirements are accurately reached with the result for R in (20). The users' rates were obtained under the assumption that , a high load scenario that constitutes an upper bound. It has been checked in the simulations that the measured probability of a nonempty queue is very close to one in our simulations.
5.2. Correlated Channel
The conclusions from the uncorrelated channel apply also here: the simulation results are satisfactory, approaching accurately the analytical results.
In this paper, we have obtained analytical expressions of the achievable users' rates in a wireless system under the conditions stated by the MAC layer: a selected scheduling discipline and a QoS constraint given in terms of a delay constraint and a BER. The delay constraint consists of a target delay and the probability of exceeding it, . Three simple and widely employed disciplines have been analyzed: Round Robin, Best Channel and Proportional Fair. The method to calculate these rates is based on the effective bandwidth theory. The analysis is done first for an uncorrelated channel and later for a time-correlated channel. The evaluation of the individual rates and the total capacity confirms the expected qualitative behaviour of the three algorithms. It is also observed that the correlation is harmful for the delay, as expected, and the maximum achievable rates decrease as the correlation increases. Moreover, the differences among users become more noticeable for more correlated channels. Finally, simulations of the algorithms were conducted to validate our outcomes.
This work has been partially supported by the Spanish Government and the European Union (Project TEC2007-67289) and the Andalusian Government (Project TIC-03226).
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