We study the problem of assigning @math identical servers to a set of @math parallel queues in a time-slotted queueing system. The connectivity of each queue to each server is randomly changing with time; each server can serve at most one queue and each queue can be served by at most one server during each time slot. Such a queueing model has been used in addressing resource allocation problems in wireless networks. It has been previously proven that Maximum Weighted Matching (MWM) is a throughput-optimal server assignment policy for such a queueing system. In this paper, we prove that for a system with i.i.d. Bernoulli packet arrivals and connectivities, MWM minimizes, in stochastic ordering sense, a broad range of cost functions of the queue lengths such as total queue occupancy (which implies minimization of average queueing delays). Then, we extend the model by considering imperfect services where it is assumed that the service of a scheduled packet fails randomly with a certain probability. We prove that the same policy is still optimal for the extended model. We finally show that the results are still valid for more general connectivity and arrival processes which follow conditional permutation invariant distributions.
For more information on optimal scheduling and resource allocation problems in wireless networks the reader is encouraged to also consult with @cite_7 @cite_28 @cite_14 @cite_19 @cite_21 @cite_24 .
@cite_14 @cite_7 @cite_28 @cite_21 @cite_24 @cite_19
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This tutorial paper overviews recent developments in optimization-based approaches for resource allocation problems in wireless systems. We begin by overviewing important results in the area of opportunistic (channel-aware) scheduling for cellular (single-hop) networks, where easily implementable myopic policies are shown to optimize system performance. We then describe key lessons learned and the main obstacles in extending the work to general resource allocation problems for multihop wireless networks. Towards this end, we show that a clean-slate optimization-based approach to the multihop resource allocation problem naturally results in a "loosely coupled" cross-layer solution. That is, the algorithms obtained map to different layers [transport, network, and medium access control physical (MAC PHY)] of the protocol stack, and are coupled through a limited amount of information being passed back and forth. It turns out that the optimal scheduling component at the MAC layer is very complex, and thus needs simpler (potentially imperfect) distributed solutions. We demonstrate how to use imperfect scheduling in the cross-layer framework and describe recently developed distributed algorithms along these lines. We conclude by describing a set of open research problems We analyze the sum capacity of multicell wireless networks with full resource reuse and channel-driven opportunistic scheduling in each cell. We address the problem of finding the co-channel (throughout the network) user assignment that results in the optimal joint multicell capacity, under a resource-fair constraint and a standard power control strategy. This problem in principle requires processing the complete co-channel gain information, and thus, has so far been justly considered unpractical due to complexity and channel gain signaling overhead. However, we expose here the following key result: The multicell optimal user scheduling problem admits a remarkably simple and fully distributed solution for large networks. This result is proved analytically for an idealized network. From this constructive proof, we propose a practical algorithm that is shown to achieve near maximum capacity for realistic cases of simulated networks of even small sizes. A challenging problem in multi-band multi-cell self-organized wireless systems, such as multi-channel Wi-Fi networks, femto pico cells in 3G 4G cellular networks, and more recent wireless networks over TV white spaces, is of distributed resource allocation. This involves four components: channel selection, client association, channel access, and client scheduling. In this paper, we present a unified framework for jointly addressing the four components with the global system objective of maximizing the clients throughput in a proportionally fair manner. Our formulation allows a natural dissociation of the problem into two sub-parts. We show that the first part, involving channel access and client scheduling, is convex and derive a distributed adaptation procedure for achieving Pareto-optimal solution. For the second part, involving channel selection and client association, we develop a Gibbs-sampler based approach for local adaptation to achieve the global objective, as well as derive fast greedy algorithms from it that achieve good solutions. We consider wireless networks with a special type of spectral allocation, where the only constraint is that a node cannot transmit to more than one receiver at a time and cannot receive more than one transmission at a time. We introduce a scheduling algorithm called regulated maximal matching which is fully distributed and guarantees a throughput that is at least half of the throughput achievable by a centralized algorithm.