arXiv subjects

Alexander Stolyar

Publications and source records attributed to Alexander Stolyar.

30 records · Page 2Linked to original sources

arXiv2016

Pull-based load distribution among heterogeneous parallel servers: the case of multiple routers

The model is a service system, consisting of several large server pools. A server processing speed and buffer size (which may be finite or infinite) depend on the pool. The input flow of customers is split equally among a fixed number of routers, which must assign customers to the servers immediately upon arrival. We consider an asymptotic regime in which the customer total arrival rate and pool sizes scale to infinity simultaneously, in proportion to a scaling parameter $n$, while the number of routers remains fixed. We define and study a multi-router generalization of the pull-based customer assignment (routing) algorithm PULL, introduced in [11] for the single-router model. Under PULL algorithm, when a server becomes idle it send a "pull-message" to a randomly uniformly selected router; each router operates independently -- it assigns an arriving customer to a server according to a randomly uniformly chosen available (at this router) pull-message, if there is any, or to a randomly uniformly selected server in the entire system, otherwise. Under Markov assumptions (Poisson arrival process and independent exponentially distributed service requirements), and under sub-critical system load, we prove asymptotic optimality of PULL: as $n\to\infty$, the steady-state probability of an arriving customer experiencing blocking or waiting, vanishes. Furthermore, PULL has an extremely low router-server message exchange rate of one message per customer. These results generalize some of the single-router results in [11].

math.PR↗

arXiv2016

Large-scale heterogeneous service systems with general packing constraints

A service system with multiple types of customers, arriving according to Poisson processes, is considered. The system is heterogeneous in that the servers also can be of multiple types. Each customer has an independent exponentially distributed service time, with the mean determined by its type. Multiple customers (possibly of different types) can be placed for service into one server, subject to "packing" constraints, which depend on the server type. Service times of different customers are independent, even if served simultaneously by the same server. The large-scale asymptotic regime is considered such that the customer arrival rates grow to infinity. We consider two variants of the model. For the {\em infinite-server} model, we prove asymptotic optimality of the {\em Greedy Random} (GRAND) algorithm in the sense of minimizing the weighted (by type) number of occupied servers in steady-state. (This version of GRAND generalizes that introduced in [15] for the homogeneous systems, with all servers of same type.) We then introduce a natural extension of GRAND algorithm for {\em finite-server} systems with blocking. Assuming subcritical system load, we prove existence, uniqueness, and local stability of the large-scale system equilibrium point such that no blocking occurs. This result strongly suggests a conjecture that the steady-state blocking probability under the algorithm vanishes in the large-scale limit.

math.PR↗

arXiv2016

A service system with randomly behaving on-demand agents

We consider a service system where agents (or, servers) are invited on-demand. Customers arrive as a Poisson process and join a customer queue. Customer service times are i.i.d. exponential. Agents' behavior is random in two respects. First, they can be invited into the system exogenously, and join the agent queue after a random time. Second, with some probability they rejoin the agent queue after a service completion, and otherwise leave the system. The objective is to design a real-time adaptive agent invitation scheme that keeps both customer and agent queues/waiting-times small. We study an adaptive scheme, which controls the number of pending agent invitations, based on queue-state feedback. We study the system process fluid limits, in the asymptotic regime where the customer arrival rate goes to infinity. The fluid limit trajectories have complicated behavior -- there are two domains where they follow different ODEs, and a "reflecting" boundary. We use the machinery of switched linear systems and common quadratic Lyapunov functions to approach the stability of fluid limits at the desired equilibrium point (with zero queues). We derive sufficient local stability conditions for the fluid limits. We conjecture that, for our model, local stability is in fact sufficient for global stability of fluid limits; the validity of this conjecture is supported by numerical and simulation experiments. When the local stability conditions do hold, simulations show good overall performance of the scheme.

math.PR↗

arXiv2015

Pull-based load distribution in large-scale heterogeneous service systems

The model is motivated by the problem of load distribution in large-scale cloud-based data processing systems. We consider a heterogeneous service system, consisting of multiple large server pools. The pools are different in that their servers may have different processing speed and/or different buffer sizes (which may be finite or infinite). We study an asymptotic regime in which the customer arrival rate and pool sizes scale to infinity simultaneously, in proportion to some scaling parameter $n$. Arriving customers are assigned to the servers by a "router", according to a {\em pull-based} algorithm, called PULL. Under the algorithm, each server sends a "pull-message" to the router, when it becomes idle; the router assigns an arriving customer to a server according to a randomly chosen available pull-message, if there are any, or to a random server, otherwise. Assuming sub-critical system load, we prove asymptotic optimality of PULL. Namely, as system scale $n\to\infty$, the steady-state probability of an arriving customer experiencing blocking or waiting, vanishes. We also describe some generalizations of the model and PULL algorithm, for which the asymptotic optimality still holds.

math.PR↗

arXiv2015

MaxWeight Scheduling: Asymptotic Behavior of Unscaled Queue-Differentials in Heavy Traffic

The model is a "generalized switch", serving multiple traffic flows in discrete time. The switch uses MaxWeight algorithm to make a service decision (scheduling choice) at each time step, which determines the probability distribution of the amount of service that will be provided. We are primarily motivated by the following question: in the heavy traffic regime, when the switch load approaches critical level, will the service processes provided to each flow remain "smooth" (i.e., without large gaps in service)? Addressing this question reduces to the analysis of the asymptotic behavior of the unscaled queue-differential process in heavy traffic. We prove that the stationary regime of this process converges to that of a positive recurrent Markov chain, whose structure we explicitly describe. This in turn implies asymptotic "smoothness" of the service processes.

math.PR↗

arXiv2014

Asymptotic optimality of a greedy randomized algorithm in a large-scale service system with general packing constraints

We consider a service system model primarily motivated by the problem of efficient assignment of virtual machines to physical host machines in a network cloud, so that the number of occupied hosts is minimized. There are multiple types of arriving customers, where a customer's mean service time depends on its type. There is an infinite number of servers. Multiple customers can be placed for service into one server, subject to general "packing" constraints. Service times of different customers are independent, even if served simultaneously by the same server. Each new arriving customer is placed for service immediately, either into a server already serving other customers (as long as packing constraints are not violated) or into an idle server. After a service completion, each customer leaves its server and the system. We propose an extremely simple and easily implementable customer placement algorithm, called Greedy-Random (GRAND). It places each arriving customer uniformly at random into either one of the already occupied servers (subject to packing constraints) or one of the so-called zero-servers, which are empty servers designated to be available to new arrivals. One instance of GRAND, called GRAND($aZ$), where $a\ge 0$ is a parameter, is such that the number of zero-servers at any given time $t$ is $aZ(t)$, where $Z(t)$ is the current total number of customers in the system. We prove that GRAND($aZ$) with $a>0$ is asymptotically optimal, as the customer arrival rates grow to infinity and $a\to 0$, in the sense of minimizing the total number of occupied servers in steady state. In addition, we study by simulations various versions of GRAND and observe the dependence of convergence speed and steady-state performance on the number of zero-servers.

math.PR↗

arXiv2014

Diffusion scale tightness of invariant distributions of a large-scale flexible service system

A large-scale service system with multiple customer classes and multiple server pools is considered, with the mean service time depending both on the customer class and server pool. The allowed activities (routing choices) form a tree (in the graph with vertices being both customer classes and server pools). We study the behavior of the system under a {\em Leaf Activity Priority} (LAP) policy, introduced in [17]. An asymptotic regime is considered, where the arrival rate of customers and number of servers in each pool tend to infinity in proportion to a scaling parameter $r$, while the overall system load remains strictly subcritical. We prove tightness of diffusion-scaled (centered at the equilibrium point and scaled down by $r^{-1/2}$) invariant distributions. As a consequence, we obtain a limit interchange result: the limit of diffusion-scaled invariant distributions is equal to the invariant distribution of the limiting diffusion process.

math.PR↗

arXiv2012

Tightness of invariant distributions of a large-scale flexible service system under a priority discipline

We consider large-scale service systems with multiple customer classes and multiple server pools; interarrival and service times are exponentially distributed, and mean service times depend both on the customer class and server pool. It is assumed that the allowed activities (routing choices) form a tree (in the graph with vertices being both customer classes and server pools). We study the behavior of the system under a Leaf Activity Priority (LAP) policy, which assigns static priorities to the activities in the order of sequential "elimination" of the tree leaves. We consider the scaling limit of the system as the arrival rate of customers and number of servers in each pool tend to infinity in proportion to a scaling parameter r, while the overall system load remains strictly subcritical. Indexing the systems by parameter r, we show that (a) the system under LAP discipline is stochastically stable for all sufficiently large r and (b) the family of the invariant distributions is tight on scales $r^{1/2 + ε}$ for all $ε> 0$. (More precisely, the sequence of invariant distributions, centered at the equilibrium point and scaled down by $r^{-(1/2 + ε)}$, is tight.)

math.PR↗

arXiv2012

An infinite server system with general packing constraints

We consider a service system model primarily motivated by the problem of efficient assignment of virtual machines to physical host machines in a network cloud, so that the number of occupied hosts is minimized. There are multiple input flows of different type customers, with a customer mean service time depending on its type. There is infinite number of servers. A server packing {\em configuration} is the vector $k=\{k_i\}$, where $k_i$ is the number of type $i$ customers the server "contains". Packing constraints must be observed, namely there is a fixed finite set of configurations $k$ that are allowed. Service times of different customers are independent; after a service completion, each customer leaves its server and the system. Each new arriving customer is placed for service immediately; it can be placed into a server already serving other customers (as long as packing constraints are not violated), or into an idle server. We consider a simple parsimonious real-time algorithm, called {\em Greedy}, which attempts to minimize the increment of the objective function $\sum_k X_k^{1+α}$, $α>0$, caused by each new assignment; here $X_k$ is the number of servers in configuration $k$. (When $α$ is small, $\sum_k X_k^{1+α}$ approximates the total number $\sum_k X_k$ of occupied servers.) Our main results show that certain versions of the Greedy algorithm are {\em asymptotically optimal}, in the sense of minimizing $\sum_k X_k^{1+α}$ in stationary regime, as the input flow rates grow to infinity. We also show that in the special case when the set of allowed configurations is determined by {\em vector-packing} constraints, Greedy algorithm can work with {\em aggregate configurations} as opposed to exact configurations $k$, thus reducing computational complexity while preserving the asymptotic optimality.

math.OC↗

arXiv2011

Multiclass multiserver queueing system in the Halfin-Whitt heavy traffic regime. Asymptotics of the stationary distribution

We consider a heterogeneous queueing system consisting of one large pool of $O(r)$ identical servers, where $r\to\infty$ is the scaling parameter. The arriving customers belong to one of several classes which determines the service times in the distributional sense. The system is heavily loaded in the Halfin-Whitt sense, namely the nominal utilization is $1-a/\sqrt{r}$ where $a>0$ is the spare capacity parameter. Our goal is to obtain bounds on the steady state performance metrics such as the number of customers waiting in the queue $Q^r(\infty)$. While there is a rich literature on deriving process level (transient) scaling limits for such systems, the results for steady state are primarily limited to the single class case. This paper is the first one to address the case of heterogeneity in the steady state regime. Moreover, our results hold for any service policy which does not admit server idling when there are customers waiting in the queue. We assume that the interarrival and service times have exponential distribution, and that customers of each class may abandon while waiting in the queue at a certain rate (which may be zero). We obtain upper bounds of the form $O(\sqrt{r})$ on both $Q^r(\infty)$ and the number of idle servers. The bounds are uniform w.r.t. parameter $r$ and the service policy. In particular, we show that $\limsup_r E \exp(θr^{-1/2}Q^r(\infty))<\infty$. Therefore, the sequence $r^{-1/2}Q^r(\infty)$ is tight and has a uniform exponential tail bound. We further consider the system with strictly positive abandonment rates, and show that in this case every weak limit $\hat{Q}(\infty)$ of $r^{-1/2}Q^r(\infty)$ has a sub-Gaussian tail. Namely $E[\exp(θ(\hat{Q}(\infty))^2)]<\infty$, for some $θ>0$.

math.PR↗

arXiv2010

Novel Architectures and Algorithms for Delay Reduction in Back-pressure Scheduling and Routing

The back-pressure algorithm is a well-known throughput-optimal algorithm. However, its delay performance may be quite poor even when the traffic load is not close to network capacity due to the following two reasons. First, each node has to maintain a separate queue for each commodity in the network, and only one queue is served at a time. Second, the back-pressure routing algorithm may route some packets along very long routes. In this paper, we present solutions to address both of the above issues, and hence, improve the delay performance of the back-pressure algorithm. One of the suggested solutions also decreases the complexity of the queueing data structures to be maintained at each node.

cs.NI↗

arXiv2010

Large number of queues in tandem: Scaling properties under back-pressure algorithm

We consider a system with N unit-service-rate queues in tandem, with exogenous arrivals of rate lambda at queue 1, under a back-pressure (MaxWeight) algorithm: service at queue n is blocked unless its queue length is greater than that of next queue n+1. The question addressed is how steady-state queues scale as N goes to infinity. We show that the answer depends on whether lambda is below or above the critical value 1/4: in the former case queues remain uniformly stochastically bounded, while otherwise they grow to infinity. The problem is essentially reduced to the behavior of the system with infinite number of queues in tandem, which is studied using tools from interacting particle systems theory. In particular, the criticality of load 1/4 is closely related to the fact that this is the maximum possible flux (flow rate) of a stationary totally asymmetric simple exclusion process.

math.PR↗