arXiv subjects

Sem Borst

Publications and source records attributed to Sem Borst.

At least 19 recordsLinked to original sources

arXiv2024

Multi-dimensional state space collapse in non-complete resource pooling scenarios

The present paper establishes an explicit multi-dimensional state space collapse (SSC) for parallel-processing systems with arbitrary compatibility constraints between servers and job types. This breaks major new ground beyond the SSC results and queue length asymptotics in the literature which are largely restricted to complete resource pooling (CRP) scenarios where the steady-state queue length vector concentrates around a line in heavy traffic. The multi-dimensional SSC that we establish reveals heavy-traffic behavior which is also far more tractable than the pre-limit queue length distribution, yet exhibits a fundamentally more intricate structure than in the one-dimensional case, providing useful insight into the system dynamics. In particular, we prove that the limiting queue length vector lives in a $K$-dimensional cone of which the set of spanning vectors is random in general, capturing the delicate interplay between the various job types and servers. For a broad class of systems we provide a further simplification which shows that the collection of random cones constitutes a fixed $K$-dimensional cone, resulting in a $K$-dimensional SSC. The dimension $K$ represents the number of critically loaded subsystems, or equivalently, capacity bottlenecks in heavy-traffic, with $K=1$ corresponding to conventional CRP scenarios. Our approach leverages probability generating function (PGF) expressions for Markovian systems operating under redundancy policies.

math.PR↗

arXiv2023

Heavy Loads and Heavy Tails

The present paper is concerned with the stationary workload of queues with heavy-tailed (regularly varying) characteristics. We adopt a transform perspective to illuminate a close connection between the tail asymptotics and heavy-traffic limit in infinite-variance scenarios. This serves as a tribute to some of the pioneering results of J.W. Cohen in this domain. We specifically demonstrate that reduced-load equivalence properties established for the tail asymptotics of the workload naturally extend to the heavy-traffic limit.

math.PR↗

arXiv2023

Deep Reinforcement Learning for Traffic Light Control in Intelligent Transportation Systems

Smart traffic lights in intelligent transportation systems (ITSs) are envisioned to greatly increase traffic efficiency and reduce congestion. Deep reinforcement learning (DRL) is a promising approach to adaptively control traffic lights based on the real-time traffic situation in a road network. However, conventional methods may suffer from poor scalability. In this paper, we investigate deep reinforcement learning to control traffic lights, and both theoretical analysis and numerical experiments show that the intelligent behavior ``greenwave" (i.e., a vehicle will see a progressive cascade of green lights, and not have to brake at any intersection) emerges naturally a grid road network, which is proved to be the optimal policy in an avenue with multiple cross streets. As a first step, we use two DRL algorithms for the traffic light control problems in two scenarios. In a single road intersection, we verify that the deep Q-network (DQN) algorithm delivers a thresholding policy; and in a grid road network, we adopt the deep deterministic policy gradient (DDPG) algorithm. Secondly, numerical experiments show that the DQN algorithm delivers the optimal control, and the DDPG algorithm with passive observations has the capability to produce on its own a high-level intelligent behavior in a grid road network, namely, the ``greenwave" policy emerges. We also verify the ``greenwave" patterns in a $5 \times 10$ grid road network. Thirdly, the ``greenwave" patterns demonstrate that DRL algorithms produce favorable solutions since the ``greenwave" policy shown in experiment results is proved to be optimal in a specified traffic model (an avenue with multiple cross streets). The delivered policies both in a single road intersection and a grid road network demonstrate the scalability of DRL algorithms.

cs.LG↗

arXiv2022

Stability of a Stochastic Ring Network

In this paper we establish a necessary and sufficient stability condition for a stochastic ring network. Such networks naturally appear in a variety of applications within communication, computer, and road traffic systems. They typically involve multiple customer types and some form of priority structure to decide which customer receives service. These two system features tend to complicate the issue of identifying a stability condition, but we demonstrate how the ring topology can be leveraged to solve the problem.

math.PR↗

arXiv2021

Power-of-two sampling in redundancy systems: the impact of assignment constraints

A classical sampling strategy for load balancing policies is power-of-two, where any server pair is sampled with equal probability. This does not cover practical settings with assignment constraints which force non-uniform sampling. While intuition suggests that non-uniform sampling adversely impacts performance, this was only supported through simulations, and rigorous statements have remained elusive. Building on product-form distributions for redundancy systems, we prove the stochastic dominance of uniform sampling for a four-server system as well as arbitrary-size systems in light traffic.

math.PR↗

arXiv2021

Fork-join and redundancy systems with heavy-tailed job sizes

We investigate the tail asymptotics of the response time distribution for the cancel-on-start (c.o.s.) and cancel-on-completion (c.o.c.) variants of redundancy-$d$ scheduling and the fork-join model with heavy-tailed job sizes. We present bounds, which only differ in the pre-factor, for the tail probability of the response time in the case of the first-come first-served (FCFS) discipline. For the c.o.s. variant we restrict ourselves to redundancy-$d$ scheduling, which is a special case of the fork-join model. In particular, for regularly varying job sizes with tail index $-\nu$ the tail index of the response time for the c.o.s. variant of redundancy-$d$ equals $-\min\{d_{\mathrm{cap}}(\nu-1),\nu\}$, where $d_{\mathrm{cap}} = \min\{d,N-k\}$, $N$ is the number of servers and $k$ is the integer part of the load. This result indicates that for $d_{\mathrm{cap}} < \frac{\nu}{\nu-1}$ the waiting time component is dominant, whereas for $d_{\mathrm{cap}} > \frac{\nu}{\nu-1}$ the job size component is dominant. Thus, having $d = \lceil \min\{\frac{\nu}{\nu-1},N-k\} \rceil$ replicas is sufficient to achieve the optimal asymptotic tail behavior of the response time. For the c.o.c. variant of the fork-join($n_{\mathrm{F}},n_{\mathrm{J}}$) model the tail index of the response time, under some assumptions on the load, equals $1-\nu$ and $1-(n_{\mathrm{F}}+1-n_{\mathrm{J}})\nu$, for identical and i.i.d. replicas, respectively; here the waiting time component is always dominant.

math.PR↗

arXiv2020

Optimal Hyper-Scalable Load Balancing with a Strict Queue Limit

Load balancing plays a critical role in efficiently dispatching jobs in parallel-server systems such as cloud networks and data centers. A fundamental challenge in the design of load balancing algorithms is to achieve an optimal trade-off between delay performance and implementation overhead (e.g. communication or memory usage). This trade-off has primarily been studied so far from the angle of the amount of overhead required to achieve asymptotically optimal performance, particularly vanishing delay in large-scale systems. In contrast, in the present paper, we focus on an arbitrarily sparse communication budget, possibly well below the minimum requirement for vanishing delay, referred to as the hyper-scalable operating region. Furthermore, jobs may only be admitted when a specific limit on the queue position of the job can be guaranteed. The centerpiece of our analysis is a universal upper bound for the achievable throughput of any dispatcher-driven algorithm for a given communication budget and queue limit. We also propose a specific hyper-scalable scheme which can operate at any given message rate and enforce any given queue limit, while allowing the server states to be captured via a closed product-form network, in which servers act as customers traversing various nodes. The product-form distribution is leveraged to prove that the bound is tight and that the proposed hyper-scalable scheme is throughput-optimal in a many-server regime given the communication and queue limit constraints. Extensive simulation experiments are conducted to illustrate the results.

cs.PF↗

arXiv2020

Achievable Stability in Redundancy Systems

We consider a system with $N$ parallel servers where incoming jobs are immediately replicated to, say, $d$ servers. Each of the $N$ servers has its own queue and follows a FCFS discipline. As soon as the first job replica is completed, the remaining replicas are abandoned. We investigate the achievable stability region for a quite general workload model with different job types and heterogeneous servers, reflecting job-server affinity relations which may arise from data locality issues and soft compatibility constraints. Under the assumption that job types are known beforehand we show for New-Better-than-Used (NBU) distributed speed variations that no replication $(d=1)$ gives a strictly larger stability region than replication $(d>1)$. Strikingly, this does not depend on the underlying distribution of the intrinsic job sizes, but observing the job types is essential for this statement to hold. In case of non-observable job types we show that for New-Worse-than-Used (NWU) distributed speed variations full replication ($d=N$) gives a larger stability region than no replication $(d=1)$.

math.PR↗

arXiv2020

Heavy-Traffic Universality of Redundancy Systems with Assignment Constraints

Service systems often face task-server assignment-constraints due to skill-based routing or geographical conditions. Redundancy scheduling responds to this limited flexibility by replicating tasks to specific servers in agreement with these assignment constraints. We gain insight from product-form stationary distributions and weak local stability conditions to establish a state space collapse in heavy traffic. In this limiting regime, the parallel-server system with redundancy scheduling operates as a multi-class single-server system, achieving full resource pooling and exhibiting strong insensitivity to the underlying assignment constraints. In particular, the performance of a fully flexible (unconstrained) system can be matched even with rather strict assignment constraints.

math.PR↗

arXiv2020

Threshold-based rerouting and replication for resolving job-server affinity relations

We consider a system with several job types and two parallel server pools. Within the pools the servers are homogeneous, but across pools possibly not in the sense that the service speed of a job may depend on its type as well as the server pool. Immediately upon arrival, jobs are assigned to a server pool. This could be based on (partial) knowledge of their type, but such knowledge might not be available. Information about the job type can however be obtained while the job is in service; as the service progresses, the likelihood that the service speed of this job type is low increases, creating an incentive to execute the job on different, possibly faster, server(s). Two policies are considered: reroute the job to the other server pool, or replicate it there. We determine the effective load per server under both the rerouting and replication policy for completely unknown as well as partly known job types. We also examine the impact of these policies on the stability bound, and find that the uncertainty in job types may significantly degrade the performance. For (highly) unbalanced service speeds full replication achieves the largest stability bound while for (nearly) balanced service speeds no replication maximizes the stability bound. Finally, we discuss how the use of threshold-based policies can help improve the expected latency for completely or partly unknown job types.

cs.PF↗

arXiv2019

Crossover times in bipartite networks with activity constraints and time-varying switching rates

In this paper we study the performance of a bipartite network in which customers arrive at the nodes of the network, but not all nodes are able to serve their customers at all times. Each node can be either active or inactive, and two nodes connected by a bond cannot be active simultaneously. This situation arises in wireless random-access networks where, due to destructive interference, stations that are close to each other cannot use the same frequency band. We consider a model where the network is bipartite, the active nodes switch themselves off at rate 1, and the inactive nodes switch themselves on at a rate that depends on time and on which half of the bipartite network they are in. An inactive node cannot become active when one of the nodes it is connected to by a bond is active. The switching protocol allows the nodes to share activity among each other. In the limit as the activation rate becomes large, we compute the crossover time between the two states where one half of the network is active and the other half is inactive. This allows us to assess the overall activity of the network depending on the switching protocol. Our results make use of the metastability analysis for hard-core interacting particle models on finite bipartite graphs derived in an earlier paper. They are valid for a large class of bipartite networks, subject to certain assumptions. Proofs rely on a comparison with switching protocols that are not time-varying, through coupling techniques.

math.PR↗

arXiv2019

Stability of Redundancy Systems with Processor Sharing

We investigate the stability condition for redundancy-d systems where each of the servers follows a processor-sharing (PS) discipline. We allow for generally distributed job sizes, with possible dependence among the d replica sizes being governed by an arbitrary joint distribution. We establish that the stability condition is characterized by the expectation of the minimum of d replica sizes being less than the mean interarrival time per server. In the special case of identical replicas, the stability condition is insensitive to the job size distribution given its mean, and the stability condition is inversely proportional to the number of replicas. In the special case of i.i.d. replicas, the stability threshold decreases (increases) in the number of replicas for job size distributions that are NBU (NWU). We also discuss extensions to scenarios with heterogeneous servers.

math.PR↗

arXiv2019

Induced idleness leads to deterministic heavy traffic limits for queue-based random-access algorithms

We examine a queue-based random-access algorithm where activation and deactivation rates are adapted as functions of queue lengths. We establish its heavy traffic behavior on a complete interference graph, which turns out to be highly nonstandard in two respects: (1) the scaling depends on some parameter of the algorithm and is not the $N/N^2$ scaling usually found in functional central limit theorems; (2) the heavy traffic limit is deterministic. We discuss how this nonstandard behavior arises from the idleness induced by the distributed nature of the algorithm. In order to prove our main result, we developed a new method for obtaining a fully coupled stochastic averaging principle.

math.PR↗

arXiv2019

Hyper-Scalable JSQ with Sparse Feedback

Load balancing algorithms play a vital role in enhancing performance in data centers and cloud networks. Due to the massive size of these systems, scalability challenges, and especially the communication overhead associated with load balancing mechanisms, have emerged as major concerns. Motivated by these issues, we introduce and analyze a novel class of load balancing schemes where the various servers provide occasional queue updates to guide the load assignment. We show that the proposed schemes strongly outperform JSQ($d$) strategies with comparable communication overhead per job, and can achieve a vanishing waiting time in the many-server limit with just one message per job, just like the popular JIQ scheme. The proposed schemes are particularly geared however towards the sparse feedback regime with less than one message per job, where they outperform corresponding sparsified JIQ versions. We investigate fluid limits for synchronous updates as well as asynchronous exponential update intervals. The fixed point of the fluid limit is identified in the latter case, and used to derive the queue length distribution. We also demonstrate that in the ultra-low feedback regime the mean stationary waiting time tends to a constant in the synchronous case, but grows without bound in the asynchronous case.

math.PR↗

arXiv2018

Deep Reinforcement Learning for Intelligent Transportation Systems

Intelligent Transportation Systems (ITSs) are envisioned to play a critical role in improving traffic flow and reducing congestion, which is a pervasive issue impacting urban areas around the globe. Rapidly advancing vehicular communication and edge cloud computation technologies provide key enablers for smart traffic management. However, operating viable real-time actuation mechanisms on a practically relevant scale involves formidable challenges, e.g., policy iteration and conventional Reinforcement Learning (RL) techniques suffer from poor scalability due to state space explosion. Motivated by these issues, we explore the potential for Deep Q-Networks (DQN) to optimize traffic light control policies. As an initial benchmark, we establish that the DQN algorithms yield the "thresholding" policy in a single-intersection. Next, we examine the scalability properties of DQN algorithms and their performance in a linear network topology with several intersections along a main artery. We demonstrate that DQN algorithms produce intelligent behavior, such as the emergence of "greenwave" patterns, reflecting their ability to learn favorable traffic light actuations.

cs.LG↗

arXiv2018

Redundancy scheduling with scaled Bernoulli service requirements

Redundancy scheduling has emerged as a powerful strategy for improving response times in parallel-server systems. The key feature in redundancy scheduling is replication of a job upon arrival by dispatching replicas to different servers. Redundant copies are abandoned as soon as the first of these replicas finishes service. By creating multiple service opportunities, redundancy scheduling increases the chance of a fast response from a server that is quick to provide service, and mitigates the risk of a long delay incurred when a single selected server turns out to be slow. The diversity enabled by redundant requests has been found to strongly improve the response time performance, especially in case of highly variable service requirements. Analytical results for redundancy scheduling are unfortunately scarce however, and even the stability condition has largely remained elusive so far, except for exponentially distributed service requirements. In order to gain further insight in the role of the service requirement distribution, we explore the behavior of redundancy scheduling for scaled Bernoulli service requirements. We establish a sufficient stability condition for generally distributed service requirements and we show that, for scaled Bernoulli service requirements, this condition is also asymptotically nearly necessary. This stability condition differs drastically from the exponential case, indicating that the stability condition depends on the service requirements in a sensitive and intricate manner.

math.PR↗

arXiv2018

Transition time asymptotics of queue-based activation protocols in random-access networks

We consider networks where each node represents a server with a queue. An active node deactivates at unit rate. An inactive node activates at a rate that depends on its queue length, provided none of its neighbors is active. For complete bipartite networks, in the limit as the queues become large, we compute the average transition time between the two states where one half of the network is active and the other half is inactive. We show that the law of the transition time divided by its mean exhibits a trichotomy, depending on the activation rate functions.

math.PR↗

arXiv2017

Load Balancing in Large-Scale Systems with Multiple Dispatchers

Load balancing algorithms play a crucial role in delivering robust application performance in data centers and cloud networks. Recently, strong interest has emerged in Join-the-Idle-Queue (JIQ) algorithms, which rely on tokens issued by idle servers in dispatching tasks and outperform power-of-$d$ policies. Specifically, JIQ strategies involve minimal information exchange, and yet achieve zero blocking and wait in the many-server limit. The latter property prevails in a multiple-dispatcher scenario when the loads are strictly equal among dispatchers. For various reasons it is not uncommon however for skewed load patterns to occur. We leverage product-form representations and fluid limits to establish that the blocking and wait then no longer vanish, even for arbitrarily low overall load. Remarkably, it is the least-loaded dispatcher that throttles tokens and leaves idle servers stranded, thus acting as bottleneck. Motivated by the above issues, we introduce two enhancements of the ordinary JIQ scheme where tokens are either distributed non-uniformly or occasionally exchanged among the various dispatchers. We prove that these extensions can achieve zero blocking and wait in the many-server limit, for any subcritical overall load and arbitrarily skewed load profiles. Extensive simulation experiments demonstrate that the asymptotic results are highly accurate, even for moderately sized systems.

math.PR↗