- Research Article
- Open Access
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 .
- Round Robin
- Delay Constraint
- Multiuser Diversity
- Violation Probability
- Target Delay
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.
2.1. 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.
Each incoming user traffic has an instantaneous rate . On his side, the wireless channel transmits at an instantaneous rate , that is, every symbol , bits can be transmitted by the channel.
Notice that in the sum above only one of the terms is nonzero, corresponding to the user allocated to the channel.
The queue size is assumed to be infinite and denotes the length of the queue at time . The dynamics of the queueing system seen by user is characterized by the equation , with .
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 is the average Signal-to-Noise Ratio of user .
where is the average energy per symbol and is the noise power spectral density.
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.
where means that and , known as the QoS exponent, is the solution to
where and are the effective bandwidth functions of the source process and the channel process for the th user, respectively.
The procedure to generalize the results to other traffic sources can be found in .
where the probability of exceeding the target delay is denoted by .
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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