Optimal control of preemptive systems with loss

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Abstract

We consider the problem of optimal preemption control in preemptive systems with loss. Based on a designed cost function composed by the following criteria: blocking cost function, preemption cost function, degradation cost function, and processing and signaling load cost function; we use the semi-Markov decision process framework as well as the value iteration algorithm to get the optimal policies. To evaluate the optimal policies, we outline their structures and the system performance for different configurations. An interesting result happens when the lower priority service becomes profitable. In this case, the performance of higher priority calls, which have the right to preempt, may be degraded. This is against the well known traffic engineering, which is solely concentrated on the resource guarantee characteristic of the preemptive priority that always improves the higher priority call performance by lowering its blocking probability.

Graphical abstract

The paper deals with the design of an optimal controller in a communication link with multiple service classes, which can come from different traffic sources. The optimal design consists of choosing one of the following actions: Acceptation (A), Acceptation with Preemption (AP) or Blocking (B) a higher priority call (HPC) service request. When an action AP is chosen, the lower priority call (LPC) is disconnected from the system (event -D).

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Highlights

► We model preemptive systems with loss by means of Semi-Markov Decision Process (SMDP) framework. ► A new cost function for evaluating the performance of preemptive systems with loss is proposed. ► Results indicate that higher priority calls QoS may be degraded when lower priority calls become more profitable.

Introduction

The fast development of new applications and deployment of new services have overloaded more and more the communication links in telecommunication networks. With multiple traffic classes inherently distinct in terms of quality of service (QoS) requirements sharing the same communication link, an adequate resource allocation scheme is mandatory to successfully handle the QoS needs of each service class as well as increase the communication link utilization. In this context, the preemption operation plays a crucial role. In the traditional traffic engineering, it has been used to ensure higher priority calls access to network resources as soon as they arrive into the system regardless of lower priority calls status [1]. This way, the blocking probability of a higher priority call is reduced. In its turn, the preempted lower priority call may be either queued until its service completion or dropped, i.e., removed from the system. Preemptive systems are classified accordingly the destination of the preempted lower priority call. In the first case in which a queue is used to accommodate the call, we have the preemptive system with delay. The latter case in which the call is disconnected from the system, we have the preemptive system with loss. Preemptive systems with loss may still be dichotomized in single link and multiple links (or multiple loss like classified in [2]). Single link systems correspond to communication systems such as last mile solutions like xDSL access, WiMax access, and cellular mobile network access. Multiple links systems correspond to networks systems such as the ones investigated in [2]. Although the relevance of preemptive systems with delay and multiple link systems in telecommunication networks, this paper focuses on preemptive systems with loss and with a single link.

In a nutshell, in a preemptive system with loss, when there is a competition for resources in a communication link, the following events take place: (i) when an incoming higher priority call arrives, the system releases an amount of resources used by ongoing lower priority calls (more than one lower priority call may be involved in this operation) and allocates to serve the higher priority call; (ii) as a consequence, the lower priority calls QoS are degraded and/or some lower priority calls are dropped; (iii) finally, as a resource allocation mechanism, the preemption operation requires an additional processing and signaling load to run and manager its execution.

Traditionally, the design of preemptive systems with loss has only considered the event i and part of the event ii, i.e., the dropping of lower priority calls. For example, see the works in references [3], [4], [5], which illustrate the application of the preemptive priority in wireless systems or more recently its application in spectrum renting also in wireless networks [6]. Suboptimal and optimal preemption policies have also been investigated in the literature. Again, the emphasis has been on the events i and part of the event ii. See for example the work of Garay and Gopal in [7], which approximates the optimal preemption policies by heuristic procedures in a centralized communication network in which the goal is to minimize the number of preempted lower priority calls or the amount of preempted bandwidth. Decentralized communication networks are studied in [8]. In this work the objective is to determine the minimum number of preempted connections by developing two algorithms mim_conn and min_bw. The preemption operation optimization has been analyzed in [9] where it is studied in the context of multiprotocol label switching (MPLS) technology in which a lower priority LSP is removed from a given path to accommodate another LSP with a higher priority. The authors proposed an integer optimization approach to minimize rerouting and a heuristic procedure to approximate the optimal result for large networks and large number of LSPs. The event iii has been neglected in all papers above mentioned.

Given the endless amount of single link communication systems with loss around the world, being deployed or in operation, and the strategic application of the preemptive priority in the design of such systems, this paper proposes and has with main contribution an investigation of the preemption operation optimization in preemptive systems with loss. Unlike the previous works, the proposed model takes into consideration the events i, ii, and iii in the system design. The optimization problem is formulated as an optimal control problem in which the goal is to find a rule for reducing the higher priority call blocking probability taking into account the set of events involved in the preemption operation such that the long run average cost per unit of time is minimal. In order to achieve this goal, the optimal preemption control problem is formulated inside the semi-Markov decision process (SMDP) framework and the value iteration algorithm is used to get the optimal policy. In the proposed cost function, together with the traditional blocking cost function used in the characterization of the admission decision of higher priority calls (event i); this paper still adopts the following cost functions: the preemption cost function, the degradation cost function, and the processing and signaling load cost function. They refer to the events ii and iii stated above. Additionally, in order to mitigate the preemption operation impact in a lower priority call, this paper uses the degradation and compensation mechanism, which allows a lower priority call to adapt its mean rate accordingly the network load. The degradation process means that accordingly the network dynamic, the bandwidth allocated to an ongoing call may be gradually reduced, while compensation consists of the reverse process [10], [11]. This way, by using more resources, a lower priority call may quickly finish its service.

The problem under analysis consists of a sequential decision problem, where the results of the actions taken may be uncertain. Sequential decision problems arise in a range of fields (inventory control, maintenance, manufacturing, and telecommunications) and have been successfully analyzed within the framework of Markov decision process (MDP) [12]. The MDP framework is an outgrowth of Markov model and dynamic programming that was developed by Bellman to study sequential decision problem in the early of 1950s. A MDP is characterized by mappings for a set of states, actions, transition probabilities, and costs (or rewards) within a process. An optimal solution seeks to minimize the sum of costs over the states under some decision policy for state-action pairs, given the update transition probabilities. In MDP, the decisions are made only in fixed epochs. However, in this work the system dynamic is governed by arrivals and departures of higher priority and lower priority calls, which are modelled by exponential distributions. This way, the times between the decision epochs are not fixed (discrete time), but random. The mathematical tool used to analyze such type of stochastic problem is the SMDP [13]. An advantage of formulating the preemption control problem as a SMDP is the possibility of successfully conciliating both treatments: optimization and performance evaluation. This is because Markov models are the natural fashion to model and evaluate the performance of communication networks [14]. Moreover, we are dealing with a dynamic system that evolves over time, so that the better way to cope with its optimization is by means of a dynamic programming approach. This holistic view (optimization and performance evaluation) is mandatory for the design of most effective resource allocation strategies in telecommunication networks.

The rest of the paper is organized as follows. Section 2 is devoted to present the system and traffic assumptions, the model formulation, the approach to solve the proposed model, and the performance metrics used to evaluate the optimal preemption policy. In Section 3, it is analyzed the optimal preemption policy performance and structure. As an important result, this paper highlights an interesting structural property of the optimal policy, which shows that when the preemption operation becomes costly, the performance of higher priority calls, which possess the preemption right, may be degraded. This is against the well known traffic engineering, which is solely concentrated on the resource guarantee characteristic of the preemptive priority that always improves the QoS perceived by higher priority calls. Finally, in Section 4, we end with the conclusions and the suggestions of forthcoming works.

Section snippets

System and traffic assumptions

We consider a single link system represented by a queuing system, without a waiting room, with C resources that are shared by two types of calls: type 1 call and type 2 call. Type 1 call, which is the higher priority call and possesses the preemption right, arrives into the system according to a Poisson process with parameter λ1 and requires negative exponential service time with mean 1/μ1. Type 2 call, which is the lower priority call, arrives into the system according to a Poisson process

Numerical and structural results

For the computation of numerical and structural results, we assume: C = 10 units of resources, 1/μ1=120S, 1/μ2=1000S,ρ1=ρ2=4,B1=1 unit of resource, [min,max]=[1,3] units of resources, cb=2. We also set the tolerance number ε=10-10. Table 1 summarizes the scenarios used in the first analysis. The goal here is to assess the impact of the preemption cost cp, the degradation cost cd, and the processing and signaling load cost co in the optimal preemption policy. To this end, the type 1 call

Conclusion

The design of resource allocation strategies in communication systems with loss makes part of the daily routine of traffic engineers. Within the many mechanisms used to improve the network performance is the preemptive priority that can be strategically employed accordingly the Service Provider objectives to leverage the higher priority calls QoS. This paper is about the optimal control of this mechanism in communication systems with loss. To this end, a SMDP approach was used to obtain the

Acknowledgment

This work is supported by the Programa de Apoio ao Doutor Recém-Contratado of the Federal University of Pará (PADRC/CAPES/FAPESPA).

Glaucio H.S. Carvalho received the B.Sc., M.Sc., and Ph.D. degrees from the Federal University of Pará (UFPA), Brazil, in 1999, 2001, and 2005, respectively, all in electrical engineering. Currently, he is an associate Professor with the Institute of Exact and Natural Sciences, Faculty of Computation at UFPA. His research interests are performance modeling and optimization of communications networks.

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    Furthermore, we are dealing with a dynamic system that evolves over time, so that the better way to cope with its optimization is by means of a dynamic programming approach. This holistic view (optimization and performance evaluation) is mandatory in the design of most effective resource allocation strategies for communication networks with multiple service classes [10]. Another aspect of our work consists of investigating how the ratio between the radius of the co-located RATs may impact on the optimal initial RAT selection.

Glaucio H.S. Carvalho received the B.Sc., M.Sc., and Ph.D. degrees from the Federal University of Pará (UFPA), Brazil, in 1999, 2001, and 2005, respectively, all in electrical engineering. Currently, he is an associate Professor with the Institute of Exact and Natural Sciences, Faculty of Computation at UFPA. His research interests are performance modeling and optimization of communications networks.

Reviews processed and approved for publication by Editor-in-Chief Dr. Manu Malek.

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