A two-class global FCFS discrete-time queueing model with arbitrary-length constant service times

We analyze a discrete-time queueing model where two types of customers, each having their own dedicated server, are accommodated in one single FCFS queue. Service times are deterministically equal to \(s \ge 1\) time slots each. New customers enter the system according to a general independent arrival process, but the types of consecutive customers may be nonindependent. As a result, arriving customers may (or may not) have the tendency to cluster according to their types, which may lead to more (or less) blocking of one type by the opposite type. The paper reveals the impact of this blocking phenomenon on the achievable throughput, the (average) system content, the (average) customer delay and the (average) unfinished work. The paper extends the results of earlier work where either the service times were assumed to be constant and equal to 1 slot each, or the customers all belonged to the same class. Our results show that, in case of Poisson arrivals, for given traffic intensity, the system-content distribution is insensitive to the length (s) of the service times, but the (mean) delay and the (mean) unfinished work in the system are not. In case of bursty arrivals, we find that all the performance measures are affected by the length (s) of the service times, for given traffic intensity.     

A multi-class discrete-time queueing system under the FCFS service discipline

The problem with the FCFS server discipline in discrete-time queueing systems is that it doesn’t actually determine what happens if multiple customers enter the system at the same time, which in the discrete-time paradigm translates into ‘during the same time-slot’. In other words, it doesn’t specify in which order such customers are served. When we consider multiple types of customers, each requiring different service time distributions, the precise order of service even starts to affect quantities such as queue content and delays of arbitrary customers, so specifying this order will be prime. In this paper we study a multi-class discrete-time queueing system with a general independent arrival process and generally distributed service times. The service discipline is FCFS and customers entering during the same time-slot are served in random order. It will be our goal to search for the steady-state distribution of queue content and delays of certain types of customers. If one thinks of the time-slot as a continuous but bounded time period, the random order of service is equivalent to FCFS if different customers have different arrival epochs within this time-slot and if the arrival epochs are independent of customer class. For this reason we propose two distinct ways of analysing; one utilizing permutations, the other considering a slot as a bounded continuous time frame.     

Effect of global FCFS and relative load distribution in two-class queues with dedicated servers

In this paper, we investigate multi-class multi-server queueing systems with global FCFS policy, i.e., where customers requiring different types of service—provided by distinct servers—are accommodated in one common FCFS queue. In such scenarios, customers of one class (i.e., requiring a given type of service) may be hindered by customers of other classes. The purpose of this paper is twofold: to gain (qualitative and quantitative) insight into the impact of (i) the global FCFS policy and (ii) the relative distribution of the load amongst the customer classes, on the system performance. We therefore develop and analyze an appropriate discrete-time queueing model with general independent arrivals, two (independent) customer classes and two class-specific servers. We study the stability of the system and derive the system-content distribution at random slot boundaries; we also obtain mean values of the system content and the customer delay, both globally and for each class individually. We then extensively compare these results with those obtained for an analogous system without global FCFS policy (i.e., with individual queues for the two servers). We demonstrate that global FCFS, as well as the relative distribution of the load over the two customer classes, may have a major impact on the system performance.     

An assignment problem for a parallel queueing system with two heterogeneous servers

In this paper, we consider an optimization problem for a parallel queueing system with two heterogeneous servers. Each server has its own queue and customers arrive at each queue according to independent Poisson processes. Each service time is independent and exponentially distributed. When a customer arrives at queue 1, the customers in queue 1 can be transferred to queue 2 by paying an assignment cost which is proportional to the number of moved customers. Holding cost is a function of the pair of queue lengths of the two servers. Our objective is to minimize the expected total discounted cost. We use the dynamic programming approach for this problem. Considering the pair of queue lengths as a state space, we show that the optimal policy has a switch over structure under some conditions on the holding cost.     

An analytic finite capacity queueing network model capturing the propagation of congestion and blocking

Analytic queueing network models often assume infinite capacity queues due to the difficulty of grasping the between-queue correlation. This correlation can help to explain the propagation of congestion. We present an analytic queueing network model which preserves the finite capacity of the queues and uses structural parameters to grasp the between-queue correlation. Unlike pre-existing models it maintains the network topology and the queue capacities exogenous. Additionally, congestion is directly modeled via a novel formulation of the state space of the queues which explicitly captures the blocking phase. The model can therefore describe the sources and effects of congestion.     

A hypercube queueing loss model with customer-dependent service rates

This paper is concerned with the solution of a specific hypercube queueing model. It extends the work that was described in a related paper by Atkinson et al. [Atkinson, J.B., Kovalenko, I.N., Kuznetsov, N., Mykhalevych, K.V., 2006. Heuristic methods for the analysis of a queuing system describing emergency medical services deployed along a highway. Cybernetics & Systems Analysis, 42, 379–391], which investigated a model for deploying emergency services along a highway. The model is based on the servicing of customer demands that arise in a number of distinct geographical zones, or atoms. Service is provided by servers that are positioned at a number of bases, each having a fixed geographical location along the highway. At each base a single server is available. Demands arising in any atom have a first-preference base and a second-preference base. If the first-preference base is busy, service is provided by the second-preference base; and, if both bases are busy, the demand is lost. In practice, because of differences in travel times from the first and second-preference bases to the atom in question, the service rate may be significantly different in the two cases. The model studied here allows for such customer-dependent service rates to occur, and the corresponding hypercube model has 3ⁿ states, where n is the number of bases. The computational intractability of this model means that exact solutions for the long-run proportion of lost demands (p_loss) can be obtained only for small values of n. In this paper, we propose two heuristic methods and a simulation approach for approximating p_loss. The heuristics are shown to produce very accurate estimates of p_loss.     

The FCFS service discipline: Stable network topologies, bounds on traffic burstiness and delay, and control by regulators

We consider the well-known First Come First Serve (FCFS) scheduling policy under bursty arrivals. This policy is commonly used in scheduling communication networks and manufacturing systems. Recently it has been shown that the FCFS policy can be unstable for some nonacyclic network topologies.We identify some network topologies under which FCFS is stable for all arrival and service rate vectors within the system's capacity. This is done by determining a sharp bound on the burstiness of traffic exiting from a tandem section of the system, in terms of the burstiness of the incoming traffic. This burstiness bound further allows us to provide a bound on the maximum number of parts in the system, and the maximum delay. It also enables us to analyze the performance of some systems controlled by the use of traffic smoothing regulators. The maximum delay can remain bounded even in the heavy traffic limit, when all stations are 100% utilized.     

A discrete-time retrial queueing model with one server

This paper presents a one-server queueing model with retrials in discrete-time. The number of primary jobs arriving in a time slot follows a general probability distribution and the different numbers of primary arrivals in consecutive time slots are mutually independent. Each job requires from the server a generally distributed number of slots for its service, and the service times of the different jobs are independent. Jobs arriving in a slot can start their service only at the beginning of the next slot. When upon arrival jobs find the server busy all incoming jobs are sent into orbit. When upon arrival in a slot jobs find the server idle, then one of the incoming jobs (randomly chosen) in that slot starts its service at the beginning of the next slot, whereas the other incoming jobs in that slot, if any, are sent into orbit. During each slot jobs in the orbit try to re-enter the system individually, independent of each other, with a given retrial probability.     

Analysis of a finite buffer model with two servers and two nonpreemptive priority classes

In this paper, we analyze a finite buffer queueing model with two servers and two nonpreemptive priority service classes. The arrival streams are independent Poisson processes, and the service times of the two classes are exponentially distributed with different means. One of the two servers is reserved exclusively for one class with high priority and the other server serves the two classes according to a nonpreemptive priority service schedule. For the model, we describe its dynamic behavior by a four-dimensional continuous-time Markov process. Applying recursive approaches we present the explicit representation for the steady-state distribution of this Markov process. Then, we calculate the Laplace–Stieltjes Transform and the steady-state distribution of the actual waiting times of two classes of customers. We also give some numerical comparison results with other queueing models.     

An M/M/2 queueing system with heterogeneous servers and multiple vacations

In this paper, a Markovian queue with two heterogeneous servers and multiple vacations has been studied. For this system, the stationary queue length distribution and mean system size have been obtained by using matrix geometric method. The busy period analysis of the system and mean waiting time distribution are discussed. Extensive numerical illustrations are provided.     

A discrete-time priority queue with two-class customers and bulk services

This paper studies the behavior of a discrete queueing system which accepts synchronized arrivals and provides synchronized services. The number of arrivals occurring at an arriving point may follow any arbitrary discrete distribution possessing finite first moment and convergent probability generating function in ¦ z ¦ 1 + with > 0. The system is equipped with an infinite buffer and one or more servers operating in synchronous mode. Service discipline may or may not be prioritized. Results such as the probability generating function of queue occupancy, average queue length, system throughput, and delay are derived in this paper. The validity of the results is also verified by computer simulations.The work reported in this paper was supported by the National Science Council of the Republic of China under Grant NSC1981-0404-E002-04.     

A queueing model for general group screening policies and dynamic item arrivals

Classification of items as good or bad can often be achieved more economically by examining the items in groups rather than individually. If the result of a group test is good, all items within it can be classified as good, whereas one or more items are bad in the opposite case. Whether it is necessary to identify the bad items or not, and if so, how, is described by the screening policy. In the course of time, a spectrum of group screening models has been studied, each including some policy. However, the majority ignores that items may arrive at random time epochs at the testing center in real life situations. This dynamic aspect leads to two decision variables: the minimum and maximum group size. In this paper, we analyze a discrete-time batch-service queueing model with a general dependency between the service time of a batch and the number of items within it. We deduce several important quantities, by which the decision variables can be optimized. In addition, we highlight that every possible screening policy can, in principle, be studied, by defining the dependency between the service time of a batch and the number of items within it appropriately.     

A queueing model with dependence between service and interarrival times

We consider a storage model that can be either interpreted as a certain queueing model with dependence between a service request and the subsequent interarrival time, or as a fluid production/inventory model with a two-state random environment. We establish a direct link between the workload distributions of the queueing model and the production/inventory model, and we present a detailed analysis of the workload and waiting time process of the queueing system.     

Consolidating or non-consolidating queues: A game theoretic queueing model with holding costs

We consider a two-server queueing system in which the servers choose their service rate based on the demand and holding cost allocation scheme offered by the demand generating entity. We provide an optimal holding cost allocation scheme that leads to the maximum possible service rate for each of a pooled and a split system. Our results suggest that careful allocation of holding costs can create incentives that enable minimum turnaround times using a common queue.     

Blow-up and global solutions to a new integrable model with two components

We will discuss a new integrable model which describes the motion of fluid. The present work is mainly concerned with global existence and blow-up phenomena which are largely due to the application of conservation laws for this integrable equations. Moreover, a new blow-up criterion for nonperiodic case is also established via the associated potential. Some interesting examples are also given to illustrate the application of our results. The precise blow-up rate is also investigated. Finally, we will emphasize the relations of classical Camassa-Holm equation and our model by analyzing the existence of global solutions.     

Tandem queueing system with infinite and finite intermediate buffers and generalized phase-type service time distribution

A tandem queueing system with infinite and finite intermediate buffers, heterogeneous customers and generalized phase-type service time distribution at the second stage is investigated. The first stage of the tandem has a finite number of servers without buffer. The second stage consists of an infinite and a finite buffers and a finite number of servers. The arrival flow of customers is described by a Marked Markovian arrival process. Type 1 customers arrive to the first stage while type 2 customers arrive to the second stage directly. The service time at the first stage has an exponential distribution. The service times of type 1 and type 2 customers at the second stage have a phase-type distribution with different parameters. During a waiting period in the intermediate buffer, type 1 customers can be impatient and leave the system. The ergodicity condition and the steady-state distribution of the system states are analyzed. Some key performance measures are calculated. The Laplace–Stieltjes transform of the sojourn time distribution of type 2 customers is derived. Numerical examples are presented.     

A queueing model for customer rescheduling and no-shows in service systems

We study an M/M/1/K queue where customers can reschedule their appointments. Rescheduled customers show up with higher probabilities, incurring lower no-show costs, but rescheduling also frees up slots that may not be filled later, leading to wasted service capacity and lower throughput. The system manager aims to minimize the long-run average cost by controlling rescheduling policies. We derive conditions under which rescheduling should be allowed for different scenarios depending on whether customers can reschedule only once or multiple times.     

An extended queueing control model for facilities with front room and back room operations and mixed-skilled workers

Three related problems arising in retail facilities with front room and back room operations, and cross-trained workers, are considered. In the first problem, the goal is to determine an optimal policy for switching workers between the two rooms. In the second and third problems, the goal is to determine the minimum number of workers that should be hired by the facility such that a satisfactory switching policy exists. We analyze two questions arising from previous work on these problems. Firstly, we demonstrate that the previously proposed policy definition is sub-optimal and cannot represent a set of solutions that are reasonable in practice, and propose an alternative definition. Secondly, we examine the problem of finding the optimal combination of cross-trained and specialized workers under a complete set of assumptions regarding worker costs. This analysis shows that the structure of the optimal solution depends on the policy definition that is employed. Moreover, under some assumptions the cost of the ‘optimal’ solution with the original policy definition will be greater than the cost of the optimal solution with the new one.     

A survey of product form queueing networks with blocking and their equivalences

Queueing network models have been extensively used to represent and analyze resource sharing systems, such as production, communication and information systems. Queueing networks with blocking are used to represent systems with finite capacity resources and with resource constraints. Different blocking mechanisms have been defined and analyzed in the literature to represent distinct behaviors of real systems with limited resources. Exact product form solutions of queueing networks with blocking have been derived, under special constraints, for different blocking mechanisms. In this paper we present a survey of product form solutions of queueing networks with blocking and equivalence properties among different blocking network models. By using such equivalences we can extend product form solutions to queueing network models with different blocking mechanisms. The equivalence properties include relationships between open and closed product form queueing networks with different blocking mechanisms.This work has been partially supported by CNR Project Research Funds Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo and by MURST Project Research Funds Performability hw/sw di sistemi distribuiti e paralleli.