On Occupancy Based Randomized Load Balancing for Large Systems with General Distributions

Multi-server architectures are ubiquitous in today's information infrastructure whether for supporting cloud services, web servers, or for distributed storage. The performance of multi-server systems is highly dependent on the load distribution. This is affected by the use of load balancing strategies. Since both latency and blocking are important features, it is most reasonable to route an incoming job to a server that is lightly loaded. Hence a good load balancing policy should be dependent on the states of servers. Since obtaining information about the remaining workload of servers for every arrival is very hard, it is preferable to design load balancing policies that depend on occupancy or the number of progressing jobs of servers. Furthermore, if the system has a large number of servers, it is not practical to use the occupancy information of all the servers to dispatch or route an arrival due to high communication cost. In large-scale systems that have tens of thousands of servers, the policies which use the occupancy information of only a finite number of randomly selected servers to dispatch an arrival result in lower implementation cost than the policies which use the occupancy information of all the servers. Such policies are referred to as occupancy based randomized load balancing policies. Motivated by cloud computing systems and web-server farms, we study two types of models. In the first model, each server is an Erlang loss server, and this model is an abstraction of Infrastructure-as-a-Service (IaaS) clouds. The second model we consider is one with processor sharing servers that is an abstraction of web-server farms which serve requests in a round-robin manner with small time granularity. The performance criterion for web-servers is the response time or the latency for the request to be processed. In most prior works, the analysis of these models was restricted to the case of exponential job length distributions and in this dissertation we study the case of general job length distributions. To analyze the impact of a load balancing policy, we need to develop models for the system's dynamics. In this dissertation, we show that one can construct useful Markovian models. For occupancy based randomized routing policies, due to complex inter-dependencies between servers, an exact analysis is mostly intractable. However, we show that the multi-server systems that have an occupancy based randomized load balancing policy are examples of weakly interacting particle systems. In these systems, servers are interacting particles whose states lie in an uncountable state space. We develop a mean-field analysis to understand a server's behavior as the number of servers becomes large. We show that under certain assumptions, as the number of servers increases, the sequence of empirical measure-valued Markov processes which model the systems' dynamics converges to a deterministic measure-valued process referred to as the mean-field limit. We observe that the mean-field equations correspond to the dynamics of the distribution of a non-linear Markov process. A consequence of having the mean-field limit is that under minor and natural assumptions on the initial states of servers, any finite set of servers can be shown to be independent of each other as the number of servers goes to infinity. Furthermore, the mean-field limit approximates each server's distribution in the transient regime when the number of servers is large. A salient feature of loss and processor sharing systems in the setting where their time evolution can be modeled by reversible Markov processes is that their stationary occupancy distribution is insensitive to the type of job length distribution; it depends only on the average job length but not on the type of the distribution. This property does not hold when the number of servers is finite in our context due to lack of reversibility. We show however that the fixed-point of the mean-field is insensitive to the job length distributions for all occupancy based randomized load balancing policies when the fixed-point is unique for job lengths that have exponential distributions. We also provide some deeper insights into the relationship between the mean-field and the distributions of servers and the empirical measure in the stationary regime. Finally, we address the accuracy of mean-field approximations in the case of loss models. To do so we establish a functional central limit theorem under the assumption that the job lengths have exponential distributions. We show that a suitably scaled fluctuation of the stochastic empirical process around the mean-field converges to an Ornstein-Uhlenbeck process. Our analysis is also valid for the Halfin-Whitt regime in which servers are critically loaded. We then exploit the functional central limit theorem to quantify the error between the actual blocking probability of the system with a large number of servers and the blocking probability obtained from the fixed-point of the mean-field. In the Halfin-Whitt regime, the error is of the order inverse square root of the number of servers. On the other hand, for a light load regime, the error is smaller than the inverse square root of the number of servers

A Continuous Variable Quantum Switch

The continuous quadratures of a single mode of the light field present a promising avenue to encode quantum information. By virtue of the infinite dimensionality of the associated Hilbert space, quantum states of these continuous variables (CV) can enable higher communication rates compared to single photon-based qubit encodings. Quantum repeater protocols that are essential to extend the range of quantum communications at enhanced rates over direct transmission have also been recently proposed for CV quantum encodings. Here we present a quantum repeating switch for CV quantum encodings that caters to multiple communication flows. The architecture of the switch is based on quantum light sources, detectors, memories, and switching fabric, and the routing protocol is based on a Max-Weight scheduling policy that is throughput optimal. We present numerical results on an achievable bipartite entanglement request rate region for multiple CV entanglement flows that can be stably supported through the switch. We elucidate our results with the help of exemplary 3-flow networks.Comment: 7 pages, 6 figures, accepted for a talk at the IEEE International Conference on Quantum Computing and Engineering (QCE), 202

A throughput optimal scheduling policy for a quantum switch

We study a quantum switch that creates shared end-to-end entangled quantum states to multiple sets of users that are connected to it. Each user is connected to the switch via an optical link across which bipartite Bell-state entangled states are generated in each time-slot with certain probabilities, and the switch merges entanglements of links to create end-to-end entanglements for users. One qubit of an entanglement of a link is stored at the switch and the other qubit of the entanglement is stored at the user corresponding to the link. Assuming that qubits of entanglements of links decipher after one time-slot, we characterize the capacity region, which is defined as the set of arrival rates of requests for end-to-end entanglements for which there exists a scheduling policy that stabilizes the switch. We propose a Max-Weight scheduling policy and show that it stabilizes the switch for all arrival rates that lie in the capacity region. We also provide numerical results to support our analysis

The mean-field behavior of processor sharing systems with general job lengths under the SQ(d) policy

This paper addresses the mean-field behavior of large-scale systems of parallel servers with a processor sharing service discipline when arrivals are Poisson and jobs have general service time distributions when an SQ() routing policy is used. Under this policy, an arrival is routed to the server with the least number of progressing jobs among randomly chosen servers. The limit of the empirical distribution is then used to study the statistical properties of the system. In particular, this shows that in the limit as grows, individual servers are statistically independent of others (propagation of chaos) and more importantly, the equilibrium point of the mean-field is insensitive to the job length distributions that has important engineering relevance for the robustness of such routing policies used in web server farms. We use a framework of measure-valued processes and martingale techniques to obtain our results. We also provide numerical results to support our analysis

Stability Analysis of a Quantum Network with Max-Weight Scheduling

We study a quantum network that distributes entangled quantum states to multiple sets of users that are connected to the network. Each user is connected to a switch of the network via a link. All the links of the network generate bipartite Bell-state entangled states in each time-slot with certain probabilities, and each end node stores one qubit of the entanglement generated by the link. To create shared entanglements for a set of users, measurement operations are performed on qubits of link-level entanglements on a set of related links, and these operations are probabilistic in nature and are successful with certain probabilities. Requests arrive to the system seeking shared entanglements for different sets of users. Each request is for the creation of shared entanglements for a fixed set of users using link-level entanglements on a fixed set of links. Requests are processed according to First-Come-First-Served service discipline and unserved requests are stored in buffers. Once a request is selected for service, measurement operations are performed on qubits of link-level entanglements on related links to create a shared entanglement. For given set of request arrival rates and link-level entanglement generation rates, we obtain necessary conditions for the stability of queues of requests. In each time-slot, the scheduler has to schedule entanglement swapping operations for different sets of users to stabilize the network. Next, we propose a Max-Weight scheduling policy and show that this policy stabilizes the network for all feasible arrival rates. We also provide numerical results to support our analysis. The analysis of a single quantum switch that creates multipartite entanglements for different sets of users is a special case of our work.Comment: 21 page

Sensitivity of mean-field fluctuations in Erlang loss models with randomized routing

In this paper, we study a large system of $N$ servers each with capacity to process at most $C$ simultaneous jobs and an incoming job is routed to a server if it has the lowest occupancy amongst $d$ (out of N) randomly selected servers. A job that is routed to a server with no vacancy is assumed to be blocked and lost. Such randomized policies are referred to JSQ(d) (Join the Shortest Queue out of $d$) policies. Under the assumption that jobs arrive according to a Poisson process with rate $N\lambda^{(N)}$ where $\lambda^{(N)}=\sigma-\frac{\beta}{\sqrt{N}}$, \sigma\in\mb{R}_+ and \beta\in\mb{R}, we establish functional central limit theorems (FCLTs) for the fluctuation process in both the transient and stationary regimes when service time distributions are exponential. In particular, we show that the limit is an Ornstein-Uhlenbeck process whose mean and variance depend on the mean-field of the considered model. Using this, we obtain approximations to the blocking probabilities for large $N$, where we can precisely estimate the accuracy of first-order approximations.Comment: 29 page

The mean-field behavior of processor sharing systems with general job lengths under the SQ(d) policy

This paper addresses the mean-field behavior of large-scale systems of parallel servers with a processor sharing service discipline when arrivals are Poisson and jobs have general service time distributions when an SQ() routing policy is used. Under this policy, an arrival is routed to the server with the least number of progressing jobs among randomly chosen servers. The limit of the empirical distribution is then used to study the statistical properties of the system. In particular, this shows that in the limit as grows, individual servers are statistically independent of others (propagation of chaos) and more importantly, the equilibrium point of the mean-field is insensitive to the job length distributions that has important engineering relevance for the robustness of such routing policies used in web server farms. We use a framework of measure-valued processes and martingale techniques to obtain our results. We also provide numerical results to support our analysis

Insensitivity of the mean field limit of loss systems under SQ(d) routeing

In this paper, we study a large multi-server loss model under the SQ(d) routeing scheme when the service time distributions are general with finite mean. Previous works have addressed the exponential service time case when the number of servers goes to infinity, giving rise to a mean field model. The fixed point of the limiting mean field equations (MFEs) was seen to be insensitive to the service time distribution in simulations, but no proof was available. While insensitivity is well known for loss systems, the models, even with state-dependent inputs, belong to the class of linear Markov models. In the context of SQ(d) routeing, the resulting model belongs to the class of nonlinear Markov processes (processes whose generator itself depends on the distribution) for which traditional arguments do not directly apply. Showing insensitivity to the general service time distributions has thus remained an open problem. Obtaining the MFEs in this case poses a challenge due to the resulting Markov description of the system being in positive orthant as opposed to a finite chain in the exponential case. In this paper, we first obtain the MFEs and then show that the MFEs have a unique fixed point that coincides with the fixed point in the exponential case, thus establishing insensitivity. The approach is via a measure-valued Markov process representation and the martingale problem to establish the mean field limit