Elsevier

Performance Evaluation

Volume 70, Issue 2, February 2013, Pages 148-159
Performance Evaluation

Stability of flow-level scheduling with Markovian time-varying channels

https://doi.org/10.1016/j.peva.2012.08.005Get rights and content

Abstract

We consider the flow-level scheduling in wireless networks. The time is slotted and in each time slot the base station selects flows/users to serve. There are multi-class users and channel conditions vary over time. The channel state for each class user is assumed to be modeled as a finite state Markov chain. Using the fluid limit approach, we find the necessary and sufficient conditions for the stability of best rate (BR) scheduling policies. As a result, we show that any BR policy is maximally stable. Our result generalizes the result of Ayesta et al. (in press) [13] and solves the conjecture of Jacko (2011) [16]. We introduce a correlated channel state model and investigate the stability condition for BR policy in this model.

Introduction

The next generation wireless networks are expected to support a wide variety of data services. Due to fading and interference effects, the wireless channel is time-varying in that the transmission rates supported for each user vary over time. The time-varying nature of wireless channels provides an opportunity to schedule flows/users when they see a favorable channel state, this is referred to as opportunistic scheduling [1], [2].

Researchers have studied scheduling in wireless systems both at the packet level and at the flow level. In packet-level models, it is typically assumed that there exists a fixed population of permanent users. We refer to [3], [4], [5] for packet-level models. In a flow-level model, users arrive randomly to the system and leave after receiving their service demands with finite size. For a review on flow-level modeling we refer to [6], [7]. The flow-level performance in general has remained largely elusive, even for basic stability properties. Therefore several studies on flow-level models were carried out under the time-scale separation assumption (see [8], [9], [10], [11]).

We consider the flow-level scheduling in wireless networks. The flows/users of random size arrive randomly at the base station, and leave when they have been completely transmitted. The time is slotted and the quality of the channel condition of each user varies per time slot. At the beginning of each time slot, the scheduler (or centralized controller) at the base station decides which users are allowed to send/receive data. Users receive/send their data from/to the controller through a commonly shared wireless channel.

Under the time-scale separation assumption, in [12], [10], the authors found the necessary and sufficient conditions for flow-level stability for a wide class of utility-based scheduling policies (including Proportional Fair (PF) scheduling and α-fair scheduling policies). We are interested in stability of Best Rate (BR) scheduling policies [13] without the time-scale separation assumption. Under a BR policy, whenever there are users that are currently in their best condition, then they will be served. The Score-Based (SB), Proportionally Best (PB) and Potential Improvement (PI) policies are included in the BR policies. An SB scheduler serves at each time slot a user whose rank of the current channel condition is best from the possible channel conditions [14]. A PB scheduler serves at each time slot a user whose ratio of the current channel condition with respect to her best possible condition is highest [8]. A PI scheduler for independent and identically distributed (i.i.d.) channel conditions was designed in [15]. A PI scheduler serves at each time slot the user with the highest ratio of the current transmission rate with respect to the expected potential improvement of the service rate. Jacko [16] considered the Markovian channel state evolution with only two possible states and introduced a new opportunistic scheduling policy, called a PI policy, as it is a generalization of a PI policy for i.i.d. time-varying channels. A PI scheduler behaves like a PI scheduler. Under i.i.d. time-varying channels PI and PI policies are identical. Remark also that a PI policy is a BR policy.

Using the fluid limit approach, Ayesta et al. [13] provided maximum stability conditions for systems with i.i.d. channel state evolution and proved that SB, PB and PI scheduling policies are maximally stable. It is likely that i.i.d. channel state evolution is unrealistic. Recently, Jacko [16] studied the optimal scheduling problem with channel state evolution governed by a two-state Markov chain.

In this paper we consider time-slotted wireless networks. There are multi-class users and channel conditions vary over time. The channel state for each class user is assumed to be modeled as a finite state Markov chain. Using the fluid limit approach, we find the necessary and sufficient conditions for the stability of BR scheduling policies. As a result, we show that any BR policy is maximally stable. Our result generalizes the result of Ayesta et al. [13] and solves the conjecture of Jacko [16]. In addition, we introduce a correlated channel state model and investigate the stability condition for BR policy in this model.

Section snippets

The model and stability results

We consider the time-slotted wireless networks with Markovian time-varying channels. The time is divided into intervals of equal length, called time slots, which are indexed by integers. There are K classes of users, and in each time slot class-k users arrive according to a batch Bernoulli process. Let akl be the probability that the number of class-k users arriving in a time slot is equal to l and let λk=l=1lakl,k=1,,K, i.e., λk is the mean number of class-k users arriving in a time slot.

Fluid limits

In this section we study fluid limits. Fluid limits have become a common tool to obtain ergodicity results (possibly in the sense of Harris) for multiclass queueing networks. The connection between the Markovian model and its associated fluid model was first established on a special example by Rybko and Stolyar [18], before Dai [19] properly defined fluid limits and generalized their approach.

For 1kK,1jmk and n=1,2,, let Ak(n) be the number of class-k users who arrive until time n. By the

An extension to a correlated channel state model

To reflect the situation of correlated channel states, we introduce a process {E(n):n=0,1,} which represents the change of environment. Let E(n) denote the environment state at time n. Assume that {E(n):n=0,1,} is a Markov process with state space {1,2} and transition probability matrix P=(ϵij)i,j=1,2. The service rate of a class-k user depends not only on its channel state but also on the environment state. That is, when a class-k user receives its service, the departure probability of a

Conclusion

We considered the flow-level scheduling in time-slotted wireless networks where there are multi-class users and the channel state of each class user evolves over time according to a finite state Markov chain. Under such a general scenario, we showed that any BR policy is maximally stable and found the stability regions for BR policies. Our result generalizes the result of Ayesta et al. [13] for i.i.d. channel state model and solves the conjecture of Jacko [16]. We introduced a correlated

Acknowledgments

The authors are grateful for the Editor’s and referee’s comments and suggestions which improved this article. The first author’s research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-0004219). The second author’s research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education,

Jeongsim Kim is an associate professor in the Department of Mathematics Education at Chungbuk National University. She received her B.S., M.S. and Ph.D. in Mathematics from Korea University. Her research interests include probability theory, queueing theory and its applications to the communication systems.

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Cited by (0)

Jeongsim Kim is an associate professor in the Department of Mathematics Education at Chungbuk National University. She received her B.S., M.S. and Ph.D. in Mathematics from Korea University. Her research interests include probability theory, queueing theory and its applications to the communication systems.

Bara Kim is a professor in the Department of Mathematics at Korea University, Seoul, Korea. He received his B.S., M.S. and Ph.D. in Mathematics from Korea Advanced Institute of Science and Technology (KAIST). His research interests include probability theory, mathematical finance, insurance models, applied operations research, queueing theory and their applications.

Jerim Kim is a Ph. D. student in the Department of Mathematics at Korea University, Seoul, Korea. She received her B.S. and M.S. in Mathematics from Korea University. Her research interests include probability theory, mathematical finance, insurance models, applied operations research, queueing theory and their applications.

Yun Han Bae received his B.S., M.S. and Ph.D. degrees in Mathematics from Korea University, Seoul, Korea, in 2003, 2005 and 2009, respectively. From September 2009 to August 2010, he was a research assistant professor at the Basic Science Institute in Korea University. From October to August 2011, he was a postdoctoral researcher at the Department of Electrical and Computer Engineering, University of Manitoba, Manitoba, Canada. From September 2011 to August 2012, he was a research assistant professor at the Telecommunication Mathematics Research center, Korea University, Seoul, Korea. He is currently an assistant professor of Department of Mathematics Education, Sangmyung University, Seoul, Korea. His research interests include queueing theory and its applications to communication systems, and performance analysis of protocols and wireless networks.

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