Kavita Ramanan

Providing quality of service over a shared wireless link

We propose an efficient way to support quality of service of multiple real-time data users sharing a wireless channel. We show how scheduling algorithms exploiting asynchronous variations of channel quality can be used to maximize the channel capacity (i.e., maximize the number of users that can be supported with the desired QoS).

SCHEDULING IN A QUEUING SYSTEM WITH ASYNCHRONOUSLY VARYING SERVICE RATES

We consider the following queuing system which arises as a model of a wireless link shared by multiple users. There is a finite number N of input flows served by a server. The system operates in discrete time t = 0,1,2,…. Each input flow can be described as an irreducible countable Markov chain; waiting customers of each flow are placed in a queue. The sequence of server states m(t), t = 0,1,2,…, is a Markov chain with finite number of states M. When the server is in state m, it can serve mim customers of flow i (in one time slot).The scheduling discipline is a rule that in each time slot chooses the flow to serve based on the server state and the state of the queues. Our main result is that a simple online scheduling discipline, Modified Largest Weighted Delay First, along with its generalizations, is throughput optimal; namely, it ensures that the queues are stable as long as the vector of average arrival rates is within the system maximum stability region.

Convex duality and the Skorokhod problem. II

Abstract. The solution to the Skorokhod Problem defines a deterministic mapping, referred to as the Skorokhod Map, that takes unconstrained paths to paths that are confined to live within a given domain G⊂ℝn. Given a set of allowed constraint directions for each point of ∂G and a path ψ, the solution to the Skorokhod Problem defines the constrained version φ of ψ, where the constraining force acts along one of the given boundary directions using the “least effort” required to keep φ in G. The Skorokhod Map is one of the main tools used in the analysis and construction of constrained deterministic and stochastic processes. When the Skorokhod Map is sufficiently regular, and in particular when it is Lipschitz continuous on path space, the study of many problems involving these constrained processes is greatly simplified.We focus on the case where the domain G is a convex polyhedron, with a constant and possibly oblique constraint direction specified on each face of G, and with a corresponding cone of constraint directions at the intersection of faces. The main results to date for problems of this type were obtained by Harrison and Reiman [22] using contraction mapping techniques. In this paper we discuss why such techniques are limited to a class of Skorokhod Problems that is a slight generalization of the class originally considered in [22]. We then consider an alternative approach to proving regularity of the Skorokhod Map developed in [13]. In this approach, Lipschitz continuity of the map is proved by showing the existence of a convex set that satisfies a set of conditions defined in terms of the data of the Skorokhod Problem. We first show how the geometric condition of [13] can be reformulated using convex duality. The reformulated condition is much easier to verify and, moreover, allows one to develop a general qualitative theory of the Skorokhod Map. An additional contribution of the paper is a new set of methods for the construction of solutions to the Skorokhod Problem.These methods are applied in the second part of this paper [17] to particular classes of Skorokhod Problems.

An explicit formula for the Skorokhod map on [0, a]

The Skorokhod map is a convenient tool for constructing solutions to stochastic differential equations with reflecting boundary conditions. In this work, an explicit formula for the Skorokhod map Γ 0,a on [0, a] for any a > 0 is derived. Specifically, it is shown that on the space D[0, ∞) of right-continuous functions with left limits taking values in R, Γ 0,a = Λ a o Γ 0 , where Λ a : D[0, ∞) → D[0, ∞) is defined by Λ a (Φ) (t) = Φ(t)- sups∈[0,t] [(Φ(s)-a) + Λ infu∈[s,t] Φ(u)] and Γ 0 :D[0, ∞) → D[0, ∞) is the Skorokhod map on [0, ∞), which is given explicitly by Γ 0 (ψ)(t) = ψ(t) + sup s∈[0,t] [-ψ(s)] + . In addition, properties of Λ a are developed and comparison properties of Γ 0,a are established.

Concentration inequalities for dependent random variables via the martingale method

The martingale method is used to establish concentration inequalities for a class of dependent random sequences on a countable state space, with the constants in the inequalities expressed in terms of certain mixing coecien ts. Along the way, bounds are obtained on mar- tingale dierences associated with the random sequences, which may be of independent interest. As applications of the main result, concentra- tion inequalities are also derived for inhomogeneous Markov chains and hidden Markov chains, and an extremal property associated with their martingale dierence bounds is established. This work complements and generalizes certain concentration inequalities obtained by Marton and Samson, while also providing a dieren t proof of some known results.

A Poisson limit for buffer overflow probabilities

A key criterion in the design of high-speed networks is the probability that the buffer content exceeds a given threshold. We consider n independent identical traffic sources modelled as point processes, which are fed into a link with speed proportional to n. Under fairly general assumptions on the input processes we show that the steady state probability of the buffer content exceeding a threshold b>0 tends to the corresponding probability assuming Poisson input processes. We verify the assumptions for a large class of long-range dependent sources commonly used to model data traffic. Our results show that with superposition, significant multiplexing gains can be achieved for even smaller buffers than suggested by previous results, which consider O(n) buffer size. Moreover, simulations show that for realistic values of the exceedance probability and moderate utilisations, convergence to the Poisson limit takes place at reasonable values of the number of sources superposed. This is particularly relevant for high-speed networks in which the cost of high-speed memory is significant.

Law of large numbers limits for many-server queues

Abstract. This work considers a many-server queueing system in which customers with i.i.d., generally distributed service times enter service in the order of arrival. The dynamics of the system are represented in terms of a process that describes the total number of customers in the system, as well as a measure-valued process that keeps track of the ages of customers in service. Under mild assumptions on the service time distribution, as the number of servers goes to infinity, a law of large numbers (or fluid) limit is established for this pair of processes. The limit is characterised as the unique solution to a coupled pair of integral equations, which admits a fairly explicit representation. As a corollary, the fluid limits of several other functionals of interest, such as the waiting time, are also obtained. Furthermore, when the arrival process is time-homogeneous, the fluid limit is shown to converge to its equilibrium. Along the way, some results of independent interest are obtained, including a continuous mapping result and a maximality property of the fluid limit. A motivation for studying these systems is that they arise as models of computer data systems and call centers.

Fluid limits of many-server queues with reneging

This work considers a many-server queueing system in which im- patient customers with i.i.d., generally distributed service times and i.i.d., generally distributed patience times enter service in the order of arrival and abandon the queue if the time before possible entry into service exceeds the patience time. The dynamics of the system is represented in terms of a pair of measure-valued processes, one that keeps track of the waiting times of the customers in queue and the other that keeps track of the amounts of time each customer being served has been in service. Under mild assumptions, essen- tially only requiring that the service and reneging distributions have densities, as the number of servers goes to innity, a law of large numbers (or uid) limit is established for this pair of processes. The limit is shown to be the unique solution of a coupled pair of deterministic integral equations that admits an explicit representation. In addition, a uid limit for the virtual waiting time process is also established. This paper extends previous work by Kaspi and Ramanan, which analyzed the model in the absence of reneging. A strong motivation for understanding performance in the presence of reneging arises from models of call centers.

SPDE limits of many-server queues.

A many-server queueing system is considered in which customers with independent and identically distributed service times enter service in the order of arrival. The state of the system is represented by a process that describes the total number of customers in the system, as well as a measure-valued process that keeps track of the ages of customers in service, leading to a Markovian description of the dynamics. Under suitable assumptions, a functional central limit theorem is established for the sequence of (centered and scaled) state processes as the number of servers goes to infinity. The limit process describing the total number in system is shown to be an Ito diffusion with a constant diffusion coefficient that is insensitive to the service distribution. The limit of the sequence of (centered and scaled) age processes is shown to be a Hilbert space valued diffusion that can also be characterized as the unique solution of a stochastic partial differential equation that is coupled with the Ito diffusion. Furthermore, the limit processes are shown to be semimartingales and to possess a strong Markov property.

A Skorokhod Problem formulation and large deviation analysis of a processor sharing model

Generalized processor sharing has been proposed as a policy for distributing processing in a fair manner between different data classes in high-speed networks. In this paper we show how recent results on the Skorokhod Problem can be used to construct and analyze the mapping that takes the input processes into the buffer content. More precisely, we show how to represent the map in terms of a Skorokhod Problem, and from this infer that the mapping is well defined (existence and uniqueness) and well behaved (Lipschitz continuity). As an elementary application we present some large deviation estimates for a many data source model.