Wireless scheduling with deadline and power constraints

Abstract

This paper studies the scheduling problem in a co-located wireless network under both the deadline and power constraints. We consider a frame-based time-slotted system. The channel condition of a link remains constant within each frame but varies from frame to frame. Packets with hard deadlines arrive at the transmitters at the beginning of each frame, and will be discarded if missing their deadlines, which are in the same frame. Each of the links is associated with a quality of service (QoS) constraint and an average transmit power constraint. A MaxWeight-type problem is formulated for achieving throughput optimality. The computational complexity of solving the MaxWeight-type problem using the exhaustive search is exponential even for a single-link system. To overcome this difficulty, we propose a greedy algorithm, named PDMax (Power and Deadline constrained MaxWeight), with complexity $O(nlog\left(n\right))$. PDMax schedules packets according to their deadlines and incremental weight gains to the objective of the MaxWeight-type problem. We prove that PDMax is throughput optimal. Our simulations further show that PDMax outperforms both the Largest-Debt-First and the greedy-MaxWeight algorithms in terms of average packet delivery ratio and average transmit power.

Introduction

The emerging applications of wireless networks in cyber–physical systems and the Internet of the Things (IoTs) require networks to support reliable real-time communications over time-varying wireless channels; and the pervasive use of battery-powered devices in these systems further requires networks to be energy efficient. Due to these emerging real-time applications of wireless networks, there has been a great interest in the development of scheduling algorithms in wireless networks to support packets with hard deadlines [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. In the seminal work [3], Hou, Borkar and Kumar proposed a frame-based framework to tackle the problem of scheduling packets with hard deadlines using a deficit counter for each data flow to measure whether the fraction of packets dropped exceeds the maximum packet dropping rate. Assuming frame-based traffic flows such that packets arrive at the beginning of each frame and the deadlines are at the end of the same frame, they proved that a low complexity scheduling algorithm, called the Largest-Debt-First (LDF), is throughput optimal in co-located networks. The frame-based framework has later been generalized in [4], where packets may arrive in the middle of a frame and the deadlines may be earlier than the end of the frame. An algorithm inspired by the MaxWeight scheduling algorithm [14] has been proposed in [4] and is proved to be throughput optimal. Besides the frame-based traffic models, a geometric approach has been introduced in [5] for general packet arrivals and deadline distributions without the frame structure, and has been used in [5], [7], [15] to quantify the efficiency ratio of the LDF. Providing end-to-end hard deadlines in multihop wireless networks has also been studied recently in [8], [10], [12], where decentralized routing and scheduling solutions have been proposed to support end-to-end hard deadlines.

Despite these significant advances on wireless scheduling with hard deadlines, only a few works simultaneously address both hard deadlines and average power constraints. For example, [2] considers a finite time horizon problem of finding the minimum energy to transmit all packets of each link and proposes the optimal policy through dynamic programming; [9] considers the problem of transmitting deadline-constrained packets over a single wireless link, and proposes an online algorithm that minimizes the transmit power; [11] considers hard deadlines and average power constraints in a wireless network with a Bernoulli packet arrival and derives the optimal policy with the Lyapunov optimization techniques; [13] proposes a near optimal scheduling and power control algorithm assuming that each link has exactly one packet to transmit in each frame, and the packet arrives at the beginning of the frame and should be delivered before the end of the frame.

The main difficulty of addressing both hard deadlines and average power constraints in wireless scheduling is the “curse of dimensionality”. As we will see, while the problem can be formulated as a MaxWeight-type problem, the computational complexity of the exhaustive search is exponential even for a single-link system. In this paper, we address this problem by developing a greedy algorithm which is provably throughput optimal. The main contributions of this paper are summarized as below:

• We consider a frame-based time-slotted $L$-link system, in which a frame consists of $T$ consecutive time slots, where $T$ is determined by the upper layer applications. Packets arrive at the beginning of each frame, and need to be delivered before their deadlines. We assume the packets that arrive at the same frame and for the same link have the same deadline, but the deadline can vary from frame to frame. We further assume the channel conditions are static within a frame and vary from frame to frame. Given such network and traffic models, we formulate an optimization problem similar to that in [4], which is a variant of the classical MaxWeight-type problem. Following the standard Lyapunov analysis, it is shown that any scheduling algorithm that solves the optimization problem is throughput optimal.

• Using exhaustive search to solve the MaxWeight-type optimization problem is computationally expensive. Even for a single-link system, the computational complexity is proved to be exponential in the summation of the deadline and the number of packets. Therefore, we propose a greedy algorithm, named PDMax. PDMax schedules packets according to their deadlines and incremental weight gains calculated by solving the optimal power control problems defined for each link. We prove that PDMax is throughput optimal, and has a computational complexity $O(LTlog\left(LT\right))$. We remark that in contrast to [3], [4], the objective function of our MaxWeight-type problem is not linear in the number of scheduled packets because the transmit power is a nonlinear function of the number of packets transmitted. Because of that, packet-by-packet greedy algorithms (such as those in [3], [4]) are no longer the right approach. The key innovation of PDMax is to map packet scheduling to time-slot scheduling where time slots are allocated to links in a greedy fashion based on the incremental weight gains. The incremental weight gains of a link are the increases of the objective function value when more time slots are allocated to the link. They are calculated by solving an optimal power control problem whose objective function again is not linear (but is convex).

• Our simulation results confirm that PDMax outperforms the greedy-MaxWeight algorithm and the LDF algorithm by achieving a higher throughput and a lower average transmit power.

The remainder of the paper is organized as follows. In Section 2, we present the system model. In Section 3, we present the formulation of the MaxWeight-type problem. In Section 4, we present the PDMax algorithm that solves the MaxWeight-type problem together with an illustrative example. In Section 5, we present the simulation results and compare the performance of PDMax with the greedy-MaxWeight and the LDF. We conclude this paper in Section 6. All proofs are included in the appendices.