Showing papers by "Ravi R. Mazumdar published in 2016"

Analysis of Randomized Join-the-Shortest-Queue (JSQ) Schemes in Large Heterogeneous Processor-Sharing Systems

Abstract: In this paper, we investigate the stability and performance of randomized dynamic routing schemes for jobs based on the Join-the-Shortest Queue (JSQ) criterion in a heterogeneous system of many parallel servers. In particular, we consider servers that use processor sharing but with different server rates, and jobs are routed to the server with the smallest occupancy among a finite number of randomly sampled servers. We focus on the case of two servers that is often referred to as a Power-of-Two scheme. We first show that in the heterogeneous setting, uniform sampling of servers can cause a loss in the stability region and thus such randomized dynamic schemes need not outperform static randomized schemes in terms of mean delay in opposition to the homogeneous case of equal server speeds where the stability region is maximal and coincides with that of the static randomized routing. We explicitly characterize the stationary distributions of the server occupancies and show that the tail distribution of the server occupancy has a super-exponential behavior as in the homogeneous case as the number of servers goes to infinity. To overcome the stability issue, we show that it is possible to combine the static state-independent scheme with a randomized JSQ scheme that allows us to recover the maximal stability region combined with the benefits of JSQ, and such a scheme is preferable in terms of average delay. The techniques are based on a mean field analysis where we show that the stationary distributions coincide with those obtained under asymptotic independence of the servers and, moreover, the stationary distributions are insensitive to the job-size distribution.

Randomized assignment of jobs to servers in heterogeneous clusters of shared servers for low delay

Abstract: We consider the problem of assignning jobs to servers in a multi-server system consisting of N parallel processor sharing servers, categorized into M (≪ N) different types according to their proces...

Binary Opinion Dynamics with Biased Agents and Agents with Different Degrees of Stubbornness

Abstract: In this paper, we investigate the impact of random interactions between agents in a social network on the diffusion of opinions in the network. Opinion of each agent is assumed to be a binary variable and each agent is assumed to be able to interact with any other agent in the network. This models scenarios where every agent in the network has to choose from two available options and the size of the neighborhood of each agent is an increasing function of the total number agents in the network. It is assumed that each agent updates its opinion at random instants upon interacting with other randomly sampled agents. We consider two simple rules of interaction: (1) the voter rule in which the updating agent simply copies the opinion of another randomly sampled agent, (2) the majority rule, in which the updating agent samples multiple agents and adopts the majority opinion among the sampled agents and the agent itself. Under each rule, we consider two different scenarios which have not been considered in the literature thus far: (1) where the agents are 'biased' towards one of the opinions, (2) where different agents have different degrees of stubbornness. We show that the presence of biased agents reduces the consensus time for the voter rule exponentially as compared to the case where the agents are unbiased. For the majority rule model with biased agents, we show that the network reaches consensus on a particular opinion with high probability only when the initial fraction of agents having that opinion is above a certain threshold. For the majority rule model with stubborn agents, we observe metastability where the network switches back and forth between stable states spending long intervals in each state.

Majority Rule Based Opinion Dynamics with Biased and Stubborn Agents

Abstract: In this paper, we investigate the impact of majority-rule based random interactions among agents in a large social network on the diffusion of opinions in the network. Opinion of each agent is assumed to be a binary variable taking values in the set {0, 1}. Interactions among agents are modeled using the majority rule, where each agent updates its opinion at random instants by adopting the 'majority' opinion among a group of randomly sampled agents. We investigate two scenarios that respectively incorporate `bias' of the agents towards a specific opinion and stubbornness of some of the agents in the majority rule dynamics. For the first scenario, where all the agents are assumed to be 'biased' towards one of the opinions, it is shown that the agents reach a consensus on the preferred opinion (with high probability) only if the initial fraction of agents having the preferred opinion is above a certain threshold. Furthermore, the mean time taken to reach the consensus is shown to be logarithmic in the network size. In the second scenario, where the presence of 'stubborn' agents, who never update their opinions, is assumed, we characterize the equilibrium distribution of opinions of the non-stubborn agents using mean field techniques. The mean field limit is shown to have multiple stable equilibrium points which leads to a phenomenon known as metastability.

Exponential Stability and the Markus-Yamabe Conjecture in Compact Spaces

Abstract: In this note we show that if a continuous-time, nonlinear, time-invariant, finite-dimensional system evolves on a compact subset of Rn and if the Jacobian of the vector field is Hurwitz at each point of the compact set, then there is a unique equilibrium on the set and solutions exponentially converge to it. This shows that the Markus-Yamabe conjecture, which is false in general on Rn, n>2, holds on compact sets. The results of this note can be viewed as an application of Krasovskii's method for constructing Lyapunov functions and we are able to similarly construct Lyapunov-like functions valid on the given compact set. Examples are provided to illustrate the result.

Effects of shadowing on the number of active users in random multi-user channels

Abstract: Prior work has addressed the effects of multipath fading and path loss separately for broadcast and multiple-access channels. It has been shown that the number of simultaneously active users are of the order $$\Theta (\ln (\ln (n)))$$ź(ln(ln(n))) for random channel gains with exponentially-decaying tails, where n is the total number of users. Furthermore, it has been shown that assuming path loss is dominant and ignoring multipath fading, the user capacity (i.e. the maximum number of simultaneously active users in a wireless system) of multi-user channels is of the order $$\Theta (\ln (n))$$ź(ln(n)) when the users are spatially distributed on the plane with a Gaussian or Uniform distribution. In this paper, we study the shadowing effects on the number of active users for broadcast and multiple-access channels in which the users are randomly distributed on the plane. It is shown that as the total number of users in the multi-user channel goes to infinity, the number of active users is of the order $$\Theta (\sqrt{\ln (n)}).$$ź(ln(n)).

Pricing schemes in processor sharing systems

Abstract: In this paper we study charging schemes for bandwidth or server usage under the processor sharing discipline. Specifically, we analyze post-payment and pre-payment (or payment on arrival) schemes in three charging frameworks: fixed-rate charging, Vickrey---Clarke---Groves based charging, and congestion based charging for users with logarithmic utilities. We show that in the absence of QoS constraints, the network operator can earn unbounded profits and thus there is a need to devise schemes where users are only charged if they are given a minimum rate. We obtain explicit characterizations for mean user payments and the operator's mean revenue for these frameworks. We also analyze charge volatility via the second moments of the above implementations of arrival-based payments and post-payments. The volatility reflects the confidence in mean revenue for the operator and expected charges for a user. We present conditions under which a pre-payment mechanism is preferable over a post-payment mechanism. We also show that the same analysis can be applied to a scenario with admission control where each entering user is guaranteed a minimum service rate.