Network Performance Optimization in Constrained Queueing Systems

Abstract

Most of literature assumed infinite buffer for users, which, however, is not practical in real networks. In this paper, we investigate downlink resource scheduling with constrained queueing. We first formulate an optimization model with the objective of maximizing the system rate under a limited queue length. Then two scheduling methods are proposed to solve this problem. One scheme is based on a virtual alarming threshold; the other is a prediction-based scheme. However, they still suffer from unfairness among rate allocations. To improve performance with respect to rate fairness, the factor of rate fairness is introduced into this optimization formulation finally. Numerical results are presented to demonstrate the efficiency of the proposed scheduling methods in terms of average system rate, maximum queue length and rate fairness, compared to some existing methods.

Appendices

Appendix A: Derivation of Lemma 1

Observed by the evolution principle of queue length based on (6), the system will be stable if the queue length is below the maximum allowed queue length after a series of scheduling. But once the number of users goes beyond a certain value, the system would be unstable even if the perfect scheduling is executed. Thus, given $Q_{max}$ and $\mu _{max}$, we need to find the upper bound of the number of users access the network, which should satisfy:

$$\begin{aligned} Q_{max}-{K\mu _{max} T}\ge 0. \end{aligned}$$

(17)

Therefore, we can get $K\le \frac{Q_{max}}{\mu _{max} T}$.

Appendix B: Derivation of Lemma 2

Given the arrival data rate for each user and the number of users $K$ access the network, there is a requirement for the minimum transmission rate for each user. If the transmission rate is below a certain level, the queue length would be not under control, leading to the unstable system. We consider the worst case here, where the queue lengths for all users will simultaneously exceed the given threshold ($G$) at the next scheduling period. Obviously, as long as the worst case is met, the system may be stable under this rate region if the suitable scheduling is employed. The current queue length ($Q_{k}(l)$) for each user is assumed as $G-\varepsilon$, where $\varepsilon$ is a small positive number. Therefore, the queue lengths of all users at the next scheduling period will exceed $G$ simultaneously. Then a simple scheduling scheme (like round-robin scheduling) is employed, where all users are scheduled in turn. In order to prevent the queue length of any user growing too fast, the queue length of the last scheduled user after $K$ scheduling periods ($Q_{k}(l+K)$) should be at least the same queue length with the original one ($Q_{k}(l)$). Otherwise, the queue length will increase without limitation, and ultimately lead to system instability. We can express the queue evolution for the last scheduled user as:

$$\begin{aligned} Q_{k}(l+K)&= Q_{k}(l)+K\mu _{max} T-r_k(l+K) T \nonumber \\&= G-\varepsilon +K\mu _{max} T-r_k(l+K) T. \end{aligned}$$

(18)

According to the analysis above, the queue length of the last scheduled user should also satisfy to maintain the system stability:

$$\begin{aligned} Q_{k}(l+K)\le Q_{k}(l). \end{aligned}$$

(19)

Through simple calculations, we can get $r_k\ge K\mu _{max}, k\in \{1,2,...,K\}$.