Analyzing the Models of Systems with Heterogeneous Servers

Distributed Computer and Communication Networks, 2020

In this paper, we propose an approximate method to investigate the Markovian queuing system with two separate pools of heterogeneous servers (HS) under N-policy. It is assumed that fast servers (F-servers) remain awake all the time while slow servers (S-servers) will go to sleep independently when number of calls in the buffer less than some threshold. At the end epoch of a sleep period, if the number of the calls gathered in the system buffer reaches or exceeds a given threshold, the corresponding S-server will wake up independently; otherwise, the S-server will begin another sleep period. An approximate method is applied under the condition that the sleep rates is essentially less than both arrival intensity of calls and their service intensity. The joint probability distribution of the number of calls in system and number of busy S-servers is determined by simple computational procedures. Illustrative numerical examples show the high accuracy of the proposed approximate method.

Mathematics, 2020

The paper studies a controllable multi-server heterogeneous queueing system where servers 1 operate at different service rates without preemption, i.e. the service times are uninterrupted. The 2 optimal control policy allocates the customers between the servers in such a way that the mean 3 number of customers in the system reaches its minimal value. The Markov decision model and the 4 policy-iteration algorithm are used to calculate the optimal allocation policy and corresponding mean 5 performance characteristics. The optimal policy, when neglecting the weak influence of slow servers, 6 is of threshold type defined as a sequence of threshold levels which specifies the queue lengths 7 for the usage of any slower server. To avoid time-consuming calculations for systems with a large 8 number of servers, we focus here on a heuristic evaluation of the optimal thresholds and compare this 9 solution with the real values. We develop also the simple lower and upper bound methods based on 10 approximation by an equivalent heterogeneous queueing system with a preemption to measure the 11 mean number of customers in the system operating under the optimal policy. Finally, the simulation 12 technique is used to provide sensitivity analysis of the heuristic solution to changes in the form of 13 inter-arrival and service time distributions. 14 Keywords: Heterogeneous servers; Markov decision process; policy-iteration algorithm; mean 15 number of customers; decomposable semi-regenerative process 16

Queueing Systems, 2016

Recent computer systems research has proposed using redundant requests to reduce latency. The idea is to run a request on multiple servers and wait for the first completion (discarding all remaining copies of the request). However, there is no exact analysis of systems with redundancy. This paper presents the first exact analysis of systems with redundancy. We allow for any number of classes of redundant requests, any number of classes of non-redundant requests, any degree of redundancy, and any number of heterogeneous servers. In all cases we derive the limiting distribution of the state of the system. In small (two or three server) systems, we derive simple forms for the distribution of response time of both the redundant classes and non-redundant classes, and we quantify the "gain" to redundant classes and "pain" to non-redundant classes caused by redundancy. We find some surprising results. First, the response time of a fully redundant class follows a simple exponential distribution and that of the non-redundant class follows a generalized hyperexponential. Second, fully redundant classes are "immune" to any pain caused by other classes becoming redundant. We also compare redundancy with other approaches for reducing latency, such as optimal probabilistic splitting of a class among servers (Opt-Split) and join-the-shortest-queue (JSQ) routing of a class. We find that, in many cases, redundancy outperforms JSQ and Opt-Split with respect to overall response time, making it an attractive solution.

Mathematics and Statistics, 2023

In this paper, we consider a single server queueing system operating in a random environment subject to disaster, repair and customer impatience. The random environment resides in any one of N + 1 phases 0, 1, 2, • • • , N + 1. The queueing system resides in phase k, k = 1, 2, • • • , N for a random interval of time and the sojourn period ends at the occurrence of a disaster. The sojourn period is exponentially distributed with mean 1/η k. At the end of the sojourn period, all customers in the system are washed out, the server goes for repair/set up and the system moves to phase 0. During the repair time, customers join the system, become impatient and leave the system. The impatience time is exponentially distributed with mean 1 ξ. Immediately after the repair, the server is ready for offering service in phase i with probability q k , k = 1, 2, • • • , N. In the k−level of the environment, customers arrive according to a Poisson process with rate λ k and the service time is exponential with mean 1/µ k. Explicit expressions for time-dependent state probabilities are found and the corresponding steady-state probabilities are deduced. Some new performance measures are also obtained. Choosing arbitrary values of the parameters subject to the stability condition, the behaviour of the system is examined. For the chosen values of the parameters, the performance measures indicated that the system did not exhibit much deviation by the presence of several phases of the environment.

Recent computer systems research has proposed using redundant requests to reduce latency. The idea is to run a single request on multiple servers and only wait for the first completion (discarding all remaining instances of the request). However no exact analysis of systems with redundancy exists. This paper presents the first exact analysis of systems with redundancy. We allow for any number of classes of redundant requests, any number of classes of non-redundant requests, any degree of redundancy, and any number of heterogeneous servers. In all cases we derive the limiting distribution on the state of the system. In small (two or three server) systems, we derive simple forms for the distribution of response time of both the redundant classes and non-redundant classes, and we quantify the "gain" to redundant classes and "pain" to non-redundant classes caused by redundancy. We find some surprising results. First, in many cases the response time of the redundant class follows a simple Exponential distribution and that of the non-redundant class follows a Generalized Hyperexponential. Second, once a class is fully redundant, it becomes "immune" to any pain caused by other classes becoming redundant. We also compare redundancy with other approaches for reducing latency, such as optimal probabilistic splitting of a class among servers (Opt-Split) and Join-the-Shortest-Queue (JSQ) routing of a class. We find that redundancy outperforms JSQ and Opt-Split with respect to overall response time, making it an attractive solution.

Industrial Engineering and Management Systems, 2016

We consider a multi-server queueing system without buffer and with two types of customers as a model of operation of a mobile network cell. Customers arrive at the system in the marked Markovian arrival flow. The service times of customers are exponentially distributed with parameters depending on the type of customer. A part of the available servers is reserved exclusively for service of first type customers. Customers who do not receive service upon arrival, can make repeated attempts. The system operation is influenced by random factors, leading to a change of the system parameters, including the total number of servers and the number of reserved servers. The behavior of the system is described by the multi-dimensional Markov chain. The generator of this Markov chain is constructed and the ergodicity condition is derived. Formulas for computation of the main performance measures of the system based on the stationary distribution of the Markov chain are derived. Numerical examples are presented.

Journal of Applied Probability, 2011

We consider several versions of the job assignment problem for an M/M/m queue with servers of different speeds. When there are two classes of customers, primary and secondary, the number of secondary customers is infinite, and idling is not permitted, we develop an intuitive proof that the optimal policy that minimizes the mean waiting time has a threshold structure. That is, for each server, there is a server-dependent threshold such that a primary customer will be assigned to that server if and only if the queue length of primary customers meets or exceeds the threshold. Our key argument can be generalized to extend the structural result to models with impatient customers, discounted waiting time, batch arrivals and services, geometrically distributed service times, and a random environment. We show how to compute the optimal thresholds, and study the impact of heterogeneity in server speeds on mean waiting times. We also apply the same machinery to the classical slow-server probl...

There are three topics in the thesis. In the first topic, we addressed a control problem for a queueing system, known as the "N-system", under the Halfin-Whitt heavy traffic regime and a static priority policy was proposed and is shown to be asymptotically optimal, using weak convergence techniques. In the second topic, we focused on the hospitals, where faster servers(nurses), though work more efficiently, have the heavier workload, and the Randomized Most-Idle (RMI) routing policy was proposed to tackle this unfairness issue, trying to reward faster servers who serve more with less workload. we extended the existing result to show that this desirable property of the RMI policy holds under a system with multiple customer classes using theoretical exact analysis as well as numerical simulations. In the third topic, the problem was to decide an appropriate number of representatives over time according to the prescribed service quality level in the call center. We examined the stability of two methods which were designed to generate appropriate staffing functions on a simulated data and real call center data from an actual bank.

Mathematics, 2020

A system with heterogeneous servers, Markov Modulated Poisson flow and instantaneous feedback is studied. The primary call is serviced on a high-speed server, and after it is serviced, each call, according to the Bernoulli scheme, either leaves the system or requires re-servicing. After the completion of servicing of a call in a slow server, according to the Bernoulli scheme, it also either leaves the system or requires re-servicing. If upon arrival of a primary call the queue length of such calls exceeds a certain threshold value and the slow server is free, then the incoming primary call, according to the Bernoulli scheme, is either sent to the slow server or joins its own queue. A mathematical model of the studied system is constructed in the form of a three-dimensional Markov chain. Approximate algorithms for calculating the steady-state probabilities of the models with finite and infinite queues are proposed and their high accuracy is shown. The results of numerical experiments...

Mathematics

The paper deals with a finite-source queueing system serving one class of customers and consisting of heterogeneous servers with unequal service intensities and of one common queue. The main model has a non-preemptive service when the customer can not change the server during its service time. The optimal allocation problem is formulated as a Markov-decision one. We show numerically that the optimal policy which minimizes the long-run average number of customers in the system has a threshold structure. We derive the matrix expressions for performance measures of the system and compare the main model with alternative simplified queuing systems which are analysed for the arbitrary number of servers. We observe that the preemptive heterogeneous model operating under a threshold policy is a good approximation for the main model by calculating the mean number of customers in the system. Moreover, using the preemptive and non-preemptive queueing models with the faster server first policy ...