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Optimal Surplus Capacity Utilization in Polling Systems Via Fluid Models 2014 12th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt) Optimal Surplus Capacity Utilization in Polling Systems via Fluid Models Ayush Rawal, Veeraruna Kavitha∗ and Manu K. Gupta Industrial Engineering and Operations Research, IIT Bombay, Powai, Mumbai - 400076, India E-mail: ayush.rawal, vkavitha, [email protected] Abstract—We discuss the idea of differential fairness in polling takes non-zero time to switch the services between the two systems. One such example scenario is: primary customers classes of customers. Another example scenario is that of data demand certain Quality of Service (QoS) and the idea is to and voice users utilizing the same wireless network. utilize the surplus server capacity to serve a secondary class of customers. We use achievable region approach for this. Towards Notion of fairness is well studied in literature (see [14], this, we consider a two queue polling system and study its [15], [17], [18] etc.,). Proportional fairness idea is introduced ‘approximate achievable region’ using a new class of delay by Kelly et. al., in [16] and a recent survey for various aspects priority kind of schedulers. We obtain this approximate region, of fairness (for work conserving queuing systems as well as via a limit polling system with fluid queues. The approximation is for polling systems) can be seen in [14]. Here they mainly accurate in the limit when the arrival rates and the service rates converge towards infinity while maintaining the load factor and discuss the fairness based on the job-size of the customer. Raz the ratio of arrival rates fixed. We show that the set of proposed et al., in [21], propose ‘Resource Allocation Queuing Fairness schedulers and the exhaustive schedulers form a complete class: Measure’ to measure fairness in allocation of system resources. every point in the region is achieved by one of those schedulers. They argued that LCFS is the least ‘fair’ service discipline It is well known that exhaustive service policy optimizes system while processor sharing is the most fair service discipline with performances like unfinished work. In this paper, we show that it is also optimal from the perspective of individual queues. respect to the measure proposed by them. We further pose two constrained optimization problems: a) ad- Fairness is achieved for under privileged users (e.g., users mission control, wherein the arrival rates of secondary customers far away from the ‘wireless’ service provider) via a constrained is optimally designed; b) maximizing the revenue considering optimization (the users were guaranteed some minimum QoS) the losses when secondary arrival rate is fixed. We finally show and is achieved using priority schedulers (see [23]). The same that exhaustive service discipline at each queue turns out to be optimal. idea is applicable even if a certain class of customers require Index Terms—Polling systems, achievable region, dynamic a QoS larger than what they would have achieved in a social scheduling, fluid queues optimal solution. We refer this idea as differential fairness, wherein, in a system with N customer classes, n classes of I. INTRODUCTION customers demand/require a certain level of Quality of Service (QoS) while the performance of the remaining N −n classes of Polling systems are special class of queueing systems where customers need to be optimized. Differential fairness concept a single server visits a set of queues in some order and could be applied in various contexts: a) application driven, for takes non-zero time to walk/switch between queues. This e.g., voice calls need to be picked within negligible waiting special class has acquired significant importance in queueing periods while data calls can tolerate delays; b) price driven, theory due to its wide range of applications in communication for e.g., certain class of customers can pay higher money to systems, production systems, traffic and transportation systems get better QoS; c) market driven, for e.g., if the plant has to [2]. Extensive research is done in both discrete and continuous manufacture different varieties of items priority is given to polling models (see [2], [3], [4], [6] etc., and references the items whose demand is more etc. It is not a completely therein). new concept, and is proposed in several other contexts (e.g., Many problems in wireless communications can be studied wireless communications) but here we are trying to generalize using polling systems. For example, consider a cognitive radio this concept to polling systems. type scenario in which a network is providing service to Achievable region approach is one of the popular techniques primary customers. We consider a slightly modified approach, to solve optimization problems (see for e.g., [7], [8], [9], [22] wherein, the primary customers are satisfied as long as their etc.,). Work conserving multi class single server queueing demands are fulfilled within a guaranteed average waiting systems pose nice geometric structure (polytope) for achiev- time. The network now can utilize the surplus capacity (the able region (e.g., [7], [22]). However structure of achievable spectrum, time slots etc.,) to serve a secondary set of users, region is not known for non work-conserving polling systems. while maintaining the QoS requirements of primary customers. Bertsimas et.al., [10] obtained the bounds on performance One can model this scenario with polling systems, when it of an optimal policy and developed optimal or near-optimal ∗This work was partially sponsored by the Indo-French Centre for Applied policies for non-work conserving polling systems, multi-class Mathematics via GANESH, an INRIA Associates program queuing systems etc. 978-3-901882-63-0/2014 - Copyright is with IFIP 381 2014 12th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt) A. Main Contribution We already have some initial observations in this direction and this is the topic of future research. We attempt to solve an example differential fairness problem (utilizing surplus capacity), via achievable region approach. B. Paper Organization A new class of parametrized schedulers are proposed, which This paper is organized as follows. Sections II ,III describe determine the switching decision between queues, and with the model and (β ; β ) schedulers. Section V presents complete motivation of finding the achievable region. These schedulers 1 2 analysis of deterministic fluid queues: we discuss stability are inspired by delay priority schedulers proposed by Klein- conditions, stationary performance and achievable region. Op- rock [19], (see also [20]) in the context of muticlass work timization problems are discussed in section VI. Some proofs conserving queuing systems. We extend these schedulers to are in Appendix while the rest are in the technical report [24]. non-work conserving, polling systems and parametrize them using β = (β ; β ): β is the parameter used when switching 1 2 i II. PROBLEM DESCRIPTION:DIFFERENTIAL FAIRNESS decision has to be made while serving at queue i. It is difficult to derive the performance of polling systems Consider a queueing system offering service to a primary with the proposed priority type of schedulers. In fact, the usual class of customers. The system is operating in stable regime, ‘load factor less than one’ is not a sufficient condition to while maintaining certain demanded QoS defined in terms of guarantee stability for such polling systems. Thus, we consider average waiting time. Operating in stable conditions implies well known fluid queues to study this problem (see for e.g., the system is under utilizing its capacity (the service rate [11], [13] etc.,). We refer these as steady state fluid model has to be less than the arrival rate). The service provider (SSFM), as these models facilitate steady state analysis and wants to utilize the surplus capacity to derive additional are useful when the system behaves periodically as in the income from secondary set of customers and this has to be example case of polling systems. By SSFM we refer two done, while maintaining the QoS of the primary customers. queue polling system with continuous, deterministic arrival This is a two class scenario in which differential fairness and departure flows. Departure flow at a queue is switched on is to be implemented and we achieve this via the following only when the server is at that queue. We analyze these fluid optimization problem: models, when switching policies are given by the proposed Optimize F(w ¯1; w¯2) Subject to: w¯1 ≤ η1; (1) (β1; β2) schedulers. We obtain stability conditions (apart from ‘load factor less than one’ we need additional condition where w¯1; w¯2 are the mean waiting times of class 1 and class 2 that β1β2 > 1) and closed form expressions of stationary customers respectively, F is an appropriate function of average performance measures for the deterministic fluid model. We waiting times and the admissible arrival rates of the the two also obtain the achievable region of waiting times which turns classes and some other relevant factors. In this paper, we out to be unbounded. consider two examples of the above problem in Section VI. We conjecture that the performance analysis of a random system (one with discrete arrivals) converges to that of the III. SYSTEM MODEL AND OUR APPROACH SSFM, as the arrival and service rates converge to infinity We consider two queue, (Q1; Q2), single server polling while maintaining certain ratios constant. Some initial ideas system. Let λi and µi be the arrival and service rates re- are discussed in technical report ([24]). Thus we have the spectively of Qi; i = 1; 2. Let S denote the random time approximate achievable region for random systems with large required by server to switch from one queue to another. The arrival and departure rates and this fact is illustrated via sequence of consecutive switching times fSkg are assumed to numerical examples.
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