CROSS-LAYER SCHEDULING DESIGN BASED ON QUEUEING THEORY

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Each link i has a queue associated with it. For simplicity, we assume that both the arrival and the departure processes of the queue are speci ed in bit level and that a source bit arrives at link i according to the Possion arrivals at a rate of li [bits per second (bps)]. When the source bit arrives at link i, it is appended to the tail of the queue. When a set of links A is activated at a timestep, the head of the queue of each link in A is removed. We have the following necessary and suf cient condition on the stability of the system. Theorem 11.1 (Condition for Stability for the Poisson Arrivals) If there is a scheduling algorithm for which the system is stable, then the rate vector of the Poisson arrival (bps) l = [l1, . . . , lK] is strictly dominated by a convex combination of the rate vectors r = [r1, . . . , rK] in the capacity region C. Conversely, if l is strictly dominated by a convex combination of the rate vectors in the capacity region C, then there exists a scheduling algorithm for which the system is stable. Proof Please refer to Appendix 11A. Theorem 11.1 provides a necessary and suf cient condition for the stability of systems with Poisson arrivals. For instance, the stability region of the system S can be written as S = l = (l1 , . . . , l K ) : l a n rn for some rn C and a n 0 n such that a n = 1 n

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(11.14)

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where the operator refers to componentwise comparison. Since Snanrn represents a convex combination of all the feasible rate vectors rn in the capacity region of the time-invariant physical layer, the stability region can be written as S = int(C ) (11.15)

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where C denotes the convex hull3 of the capacity region in the physical layer and int(.) denotes the interior of a set. Figure 11.6 illustrates an example of the capacity region of a two-user time-invariant channel as well as the corresponding stability region. Any Poisson arrival rate vector l S will make the system stable with an appropriately designed scheduler. On the other hand, for any Possion arrival rate vector l outside the stability region S, no scheduling algorithm that can make the system stable exists. The stability region is useful in the admission control of the data traf c in the cellular system. The

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The convex hull of a set of vectors is the smallest convex set that contains all the vectors.

STABILITY REGION

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4 possible rate vectors

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Stability region

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Convex hull (capacity region achieved by timesharing)

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Figure 11.6. Illustration of capacity region and stability region for a two-user time-invariant channel: (a) capacity region C of a two-user channel; (b) stability region S = int(C ).

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system should keep the the arrival rates within the stability region as otherwise the buffers in the system will over ow. 11.4.2 Stability Region of Stochastic Physical Layer

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In Section 11.4.1, we characterized the stability region for the deterministic (or time-invariant) physical layer. The stability region is strictly dominated by the convex combination of the rate vectors in the capacity region of the underlying physical layer. In the following, we shall characterize the stability region when the underlying physical layer is time-varying in a random manner. In other words, we shall focus on the stability region of the system with the stochastic multiuser physical layer. This is very important because in wireless communications, channels are randomly varying with time because different and random interference levels are observed by different users, as well as fast fading of a signal received by a user. We consider a general multiuser physical layer shared by K users where the capacity region is time-varying and is governed by an underlying random process m(t) M, which is referred to as the state of the wireless channel. The multiuser channel is characterized by a discrete channel state m M = [1, . . . , M]. The capacity region when the multiuser physical channel is at state m(t) is denoted by C(m). When the physical layer is at state m(t) and when a m m feasible rate vector rm = [r 1 , . . . , rK ] C(m) is selected by the scheduler, the m data rate of user i is given by ri (bps). Here, we allow more than one user to be selected (i.e., there exists rim > 0 and rjm > 0 for some i j). For example, if rm = [1, 2, 0, 0] is selected by the scheduler, then the instantaneous scheduled rate of users 1 and 2 are 1 and 2 bps, respectively. The scheduled data rates of

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