Ina Maria Verloop

Scheduling in a random environment: stability and asymptotic optimality

We investigate the scheduling of a common resource between several concurrent users when the feasible transmission rate of each user varies randomly over time. Time is slotted, and users arrive and depart upon service completion. This may model, for example, the flow-level behavior of end-users in a narrowband HDR wireless channel (CDMA 1xEV-DO). As performance criteria, we consider the stability of the system and the mean delay experienced by the users. Given the complexity of the problem, we investigate the fluid-scaled system, which allows to obtain important results and insights for the original system: 1) We characterize for a large class of scheduling policies the stability conditions and identify a set of maximum stable policies, giving in each time-slot preference to users being in their best possible channel condition. We find in particular that many opportunistic scheduling policies like Score-Based, Proportionally Best, or Potential Improvement are stable under the maximum stability conditions, whereas the opportunistic scheduler Relative-Best or the cμ-rule are not. 2) We show that choosing the right tie-breaking rule is crucial for the performance (e.g., average delay) as perceived by a user. We prove that a policy is asymptotically optimal if it is maximum stable and the tie-breaking rule gives priority to the user with the highest departure probability. We will refer to such tie-breaking rule as myopic. 3) We derive the growth rates of the number of users in the system in overload settings under various policies, which give additional insights on the performance. 4) We conclude that simple priority-index policies with the myopic tie-breaking rule are stable and asymptotically optimal. All our findings are validated with extensive numerical experiments.

Energy-aware capacity scaling in virtualized environments with performance guarantees

We investigate the trade-off between performance and power consumption in servers hosting virtual machines running IT services. The performance behavior of such servers is modeled through Generalized Processor Sharing (GPS) queues enhanced with a green speed-scaling mechanism that controls the processing capacity to use depending on the number of active virtual machines. When the number of virtual machines grows large, we show that the stochastic evolution of our model converges to a system of ordinary differential equations for which we derive a closed-form formula for its unique stationary point. This point is a function of the capacity and the shares that characterize the GPS mechanism. It allows us to show that speed-scaling mechanisms can provide large reduction in power consumption having only small performance degradation in terms of the delays experienced in the virtual machines. In addition, we derive the optimal choice for the shares of the GPS discipline, which turns out to be non-trivial. Finally, we show how our asymptotic analysis can be applied to the dimensioning and service partitioning in data-centers. Experimental results show that our asymptotic formulas are accurate even when the number of virtual machines is small.

Monotonicity Properties for Multi-Class Queueing Systems

We study multi-dimensional stochastic processes that arise in queueing models used in the performance evaluation of wired and wireless networks. The evolution of the stochastic process is determined by the scheduling policy used in the associated queueing network. For general arrival and service processes, we give sufficient conditions in order to compare sample-path wise the workload and the number of users under different policies. This allows us to evaluate the performance of the system under various policies in terms of stability, the mean overall delay and the mean holding cost. We apply the general framework to linear networks, where users of one class require service from several shared resources simultaneously. For the important family of weighted α-fair policies, stability results are derived and monotonicity of the mean holding cost with respect to the fairness parameter α and the relative weights is established. In order to broaden the comparison results, we investigate a heavy-traffic regime and perform numerical experiments. In addition, we study a single-server queue with two user classes, and show that under Discriminatory Processor Sharing (DPS) or Generalized Processor Sharing (GPS) the mean overall sojourn time is monotone with respect to the ratio of the weights. Finally we extend the framework to obtain comparison results that cover the single-server queue with an arbitrary number of classes as well.

Asymptotically optimal parallel resource assignment with interference

Motivated by scheduling in cellular wireless networks and resource allocation in computer systems, we study a service facility with two classes of users having heterogeneous service requirement distributions. The aggregate service capacity is assumed to be largest when both classes are served in parallel, but giving preferential treatment to one of the classes may be advantageous when aiming at minimization of the number of users, or when classes have different economic values, for example.We set out to determine the allocation policies that minimize the total number of users in the system. For some particular cases we can determine the optimal policy exactly, but in general this is not analytically feasible. We then study the optimal policies in the fluid regime, which prove to be close to optimal in the original stochastic model. These policies can be characterized by either linear or exponential switching curves. We numerically compare our results with existing approximations based on optimization in the heavy-traffic regime. By simulations we show that, in general, our simple computable switching-curve strategies based on the fluid analysis perform well.

Index policies for a multi-class queue with convex holding cost and abandonments

We investigate a resource allocation problem in a multi-class server with convex holding costs and user impatience under the average cost criterion. In general, the optimal policy has a complex dependency on all the input parameters and state information. Our main contribution is to derive index policies that can serve as heuristics and are shown to give good performance. Our index policy attributes to each class an index, which depends on the number of customers currently present in that class. The index values are obtained by solving a relaxed version of the optimal stochastic control problem and combining results from restless multi-armed bandits and queueing theory. They can be expressed as a function of the steady-state distribution probabilities of a one-dimensional birth-and-death process. For linear holding cost, the index can be calculated in closed-form and turns out to be independent of the arrival rates and the number of customers present. In the case of no abandonments and linear holding cost, our index coincides with the cμ-rule, which is known to be optimal in this simple setting. For general convex holding cost we derive properties of the index value in limiting regimes: we consider the behavior of the index (i) as the number of customers in a class grows large, which allows us to derive the asymptotic structure of the index policies, and (ii) as the abandonment rate vanishes, which allows us to retrieve an index policy proposed for the multi-class M/M/1 queue with convex holding cost and no abandonments. In fact, in a multi-server environment it follows from recent advances that the index policy is asymptotically optimal for linear holding cost. To obtain further insights into the index policy, we consider the fluid version of the relaxed problem and derive a closed-form expression for the fluid index. The latter coincides with the stochastic model in case of linear holding costs. For arbitrary convex holding cost the fluid index can be seen as the Gcμθ-rule, that is, including abandonments into the generalized cμ-rule (Gcμ-rule). Numerical experiments show that our index policies become optimal as the load in the system increases.

Dynamic fluid-based scheduling in a multi-class abandonment queue

We investigate how to share a common resource among multiple classes of customers in the presence of abandonments. We consider two different models: (1) customers can abandon both while waiting in the queue and while being served, (2) only customers that are in the queue can abandon. Given the complexity of the stochastic optimization problem we propose a fluid model as a deterministic approximation. For the overload case we directly obtain that the c@?@m/@q rule is optimal. For the underload case we use Pontryagins Maximum Principle to obtain the optimal solution for two classes of customers; there exists a switching curve that splits the two-dimensional state-space into two regions such that when the number of customers in both classes is sufficiently small the optimal policy follows the c@?@m-rule and when the number of customers is sufficiently large the optimal policy follows the c@?@m/@q-rule. The same structure is observed in the optimal policy of the stochastic model for an arbitrary number of classes. Based on this we develop a heuristic and by numerical experiments we evaluate its performance and compare it to several index policies. We observe that the suboptimality gap of our solution is small.

Sojourn times in a processor sharing queue with multiple vacations

We study an M/G/1 processor sharing queue with multiple vacations. The server only takes a vacation when the system has become empty. If he finds the system still empty upon return, he takes another vacation, and so on. Successive vacations are identically distributed, with a general distribution. When the service requirements are exponentially distributed we determine the sojourn time distribution of an arbitrary customer. We also show how the same approach can be used to determine the sojourn time distribution in an M/M/1-PS queue of a polling model, under the following constraints: the service discipline at that queue is exhaustive service, the service discipline at each of the other queues satisfies a so-called branching property, and the arrival processes at the various queues are independent Poisson processes. For a general service requirement distribution we investigate both the vacation queue and the polling model, restricting ourselves to the mean sojourn time.

Sojourn time approximations in a multi-class time-sharing server

We study a multi-class time-sharing discipline with relative priorities known as Discriminatory Processor Sharing (DPS), which provides a natural framework to model service differentiation in systems. The analysis of DPS is extremely challenging and analytical results are scarce. We develop closed-form approximations for the mean conditional and unconditional sojourn times. The main benefits of the approximations lie in its simplicity, the fact that it applies for general service requirements with finite second moments, and that it provides insights into the dependency of the performance on the system parameters. We show that the approximation for the mean (un)conditional sojourn time of a customer is decreasing as its relative priority increases. We also show that the approximation is exact in various scenarios, and that it is uniformly bounded in the second moments of the service requirements. Finally we numerically illustrate that the approximation is accurate across a broad range of parameters.

Technical vulnerability of the E-UTRAN paging mechanism

The signalling subsystem is the most expensive and complex element in cellular networks. Todays networks use signalling mechanisms whose design builds on more than 20 years of the operational expertise. Despite this, the signalling subsystem of all mobile network standards remains vulnerable to failures of equipment and to sharp increases in offered load caused by unanticipated traffic patterns. In this work we analyse the technical vulnerability of paging - a key signalling mechanism that is responsible for notifying mobile terminals of downlink service requests. Our assessment considers the operation of Long-Term Evolution (LTE) networks and in particular the performance of the paging mechanism on the radio interface. We implemented the paging procedure in a simulator, and we propose and verify a mathematical model of the system behaviour. The proposed mathematical model effectively captures the non-zero threshold of paging failure probability; it is scrupulously precise for the paging loads below the nonzero threshold and delivers a close approximation with the increasing paging load. The mathematical model can be applied to optimise the performance of the paging mechanism and to devise techniques able to overcome its technical vulnerability.

Maximal flow‐level stability of best‐rate schedulers in heterogeneous wireless systems

We investigate flow-level stability of schedulers in parallel-service wireless systems, which is important for maximizing the base stations capacity to serve the heterogeneous flows that are within the base stations power range. We model such a system as a multi-class queueing system with multiple preemptive servers, in which flows of different classes randomly arrive and depart once their flow is completed. The channel condition of a flow varies randomly over time because of fading and mobility. The evolution of the channel condition is assumed to be Markovian and class dependent. We focus on a general family of the best-rate schedulers that, whenever possible, serve flows that are in the channel condition corresponding to the highest achievable class-dependent transmission rate (i.e. the best rate). We prove under mild assumptions that any best-rate scheduler achieves maximal stability, that is, stabilizes the system whenever possible, in all systems with generally distributed class-dependent arrivals and flow sizes.