Two-moment analysis of alternative tool models with random breakdowns

A single-server queueing system with stationary Poisson arrival process, finite buffer and impatient customers as a model of TCP Short Transfer is investigated. Service time of a customer by a server has an exponential distribution. If the server is busy and the buffer is full at a customer arrival epoch, the customer may leave the system forever or move to the orbit. Customers staying in the buffer exhibit signs of impatience: they can leave the buffer lying out of the service. Patience time of a customer has an exponential distribution. When this time expires the customer either leaves the system permanently or goes to the orbit of infinite size. Customers staying in the orbit repeat their attempts to get the service later on in a random amount of time. This time is exponentially distributed with the rate depending or independent of the current number of customers in the orbit.Behavior of the system under study is described by two-dimensional asymptotically quasi-Toeplitz Markov chain. Stability conditions and the algorithms for calculating the stationary state distribution of the chain are obtained. Main performance measures of the system are calculated.KeywordsRetrial queueing systemFinite bufferImpatient customersStationary performance measures

We consider the stability of N-model systems that consist of two customer classes and two server pools. Servers in one of the pools can serve both classes, but those in the other pool can serve only one of the classes. The standard fluid models in general are not sufficient to establish the stability region of these systems under static priority policies. Therefore, we use a novel and a general approach to augment the fluid model equations based on induced Markov chains. Using this new approach, we establish the stability region of these systems under a static priority rule with thresholds when the service and interarrival times have phase-type distributions. We show that, in certain cases, the stability region depends on the distributions of the service and interarrival times (beyond their mean), on the number of servers in the system, and on the threshold value. We also show that it is possible to expand the stability region in these systems by increasing the variability of the service times (without changing their mean) while keeping the other parameters fixed. The extension of our results to parallel server systems and general service time distributions is also discussed.

The main challenge in scheduling multi-cluster tools in semiconductor manufacturing is the interactions among clusters. These interactions create a k-unit optimal production that do not exist in single-cluster tools. This paper analyzes optimal scheduling of k-unit cycle production of multi-cluster tools with single-blade robots. A resource-based method is used to analytically derive closed-form expressions for the minimal cycle time of a multi-cluster tool. Conditions for decoupling multi-cluster tools and optimality conditions for the widely used pull schedule are also presented. An example from industry production is used to illustrate the derived formula and decoupling conditions.

A single-server queueing system with a marked Markovian arrival process of heterogeneous customers is considered. Type-1 customers have limited preemptive priority over type-2 customers. There is an infinite buffer for type-2 customers and no buffer for type-1 customers. There is also a finite buffer (stock) for consumable additional items (semi-products, half-stocks, etc.) which arrive according to the Markovian arrival process. Service of a customer requires a fixed number of consumable additional items depending on the type of the customer. The service time has a phase-type distribution depending on the type of the customer. Customers in the buffer are impatient and may leave the system without service after an exponentially distributed amount of waiting time. Aiming to minimize the loss probability of type-1 customers and maximize throughput of the system, a threshold strategy of admission to service of type-2 customers is offered. Service of type-2 customer can start only if the server is idle and the number of consumable additional items in the stock exceeds the fixed threshold. Stationary distributions of the system states and the waiting time are computed. In the numerical example, we show some interesting effects and illustrate a possibility of application of the presented results for solution of optimization problems.

This paper is concerned on an M/M/1 queue with correlated interarrival and service times. In particular, we assume that the interarrival time and service time of a customer have a bivariate exponential distribution. By utilizing a Markov modulated fluid flow (MMFF) process associated with the age process of the customer in service, we obtain a number of queueing quantities in closed form. Using the solutions, the impact of the correlation coefficient of the interarrival and service times on a variety of queueing quantities is explored quantitatively and qualitatively. Specifically, we establish a monotonic relationship between the decay rates of queueing quantities and the correlation coefficient of the interarrival and service times. We also show that the decay rates are increasing in the correlation coefficient. In addition, queues with the maximum/minimum correlation coefficient are analyzed. Two examples are presented to demonstrate the importance of using the correlation coefficient in queueing performance/economic analysis.

Models are given for sequences of correlated exponential interarrival and service times for a single-server queue. These multivariate exponential models are formed as probabilistic linear combinations of sequences of independent exponential random variables and are easy to generate on a computer. Limiting results for customer waiting time under heavy traffic conditions are obtained for these queues. Heavy traffic results are useful for analyzing the effect of correlated interarrival and service times in queues on such quantities as queue length and customer waiting time. They can also be used to check simulation results.

We consider single-server queueing systems that are modulated by a discrete-time Markov chain on a countable state space. The underlying stochastic process is a Markov random walk (MRW) whose increments can be expressed as differences between service times and interarrival times. We derive the joint distributions of the waiting and idle times in the presence of the modulating Markov chain. Our approach is based on properties of the ladder sets associated with this MRW and its time-reversed counterpart. The special case of a Markov-modulated M/M/1 queueing system is then analysed and results analogous to the classical case are obtained.

We consider single-server queueing systems that are modulated by a discrete-time Markov chain on a countable state space. The underlying stochastic process is a Markov random walk (MRW) whose increments can be expressed as differences between service times and interarrival times. We derive the joint distributions of the waiting and idle times in the presence of the modulating Markov chain. Our approach is based on properties of the ladder sets associated with this MRW and its time-reversed counterpart. The special case of a Markov-modulated M/M/1 queueing system is then analysed and results analogous to the classical case are obtained.

In this work, the fluid limit approach methodology is applied to find a sufficient and necessary stability condition for the Basic Collaboration (BC) system with feedback allowed, which is a generalization of the so-called W-model. In this queueing system, some customer classes need cooperation of a subset of (non-overlapping) servers. We assume that each customer class arrives to the system following a renewal input with general i.i.d. inter-arrival times, and general i.i.d. service times are also assumed. Priority is given to customer classes that can not be served by a single server but need a cooperation.

In this paper we study a broad class of semi-Markovian queues introduced by Sengupta. This class contains many classical queues such as the GI/M/1 queue, SM/MAP/1 queue and others, as well as queues with correlated inter-arrival and service times. Queues belonging to this class are characterized by a set of matrices of size m and Sengupta showed that its waiting time distribution can be represented as a phase-type distribution of order m. For the special case of the SM/MAP/1 queue without correlated service and inter-arrival times the queue length distribution was also shown to be phase-type of order m, but no derivation for the queue length was provided in the general case. This paper introduces an order m^2 phase-type representation (kappa, K) for the queue length distribution in the general case. Moreover, we prove that the order m^2 of the distribution cannot be further reduced in general. Examples for which the order is between m and m^2 are also identified. We derive these results in both discrete and continuous time and also discuss the numerical procedure to compute (kappa, K). Moreover, by combining a result of Sengupta and Ozawa, we provide a simple formula to compute the order m phase-type representation of the waiting time in a MAP/MAP/1 queue without correlated service and inter-arrival times, using the R matrix of a Quasi-Birth-Death Markov chain.

A matrix geometric representation for the queue length distribution of multitype semi-Markovian queues

The simple model NMEAR (1) is described for a stationary dependent sequence of random variables which have a mixed exponential marginal distribution; the model is a first-order stochastic difference equation with random coefficients and is first-order Markovian. It should be broadly applicable for stochastic modelling in operations analysis. In particular, it provides a model for simulating interarrival times in queuing systems when these random variables are overdispersed relative to an exponential random variable, and moreover are positively correlated. The model also has capability to model a variable which may be zero, but which otherwise is exponentially distributed. Such variables are found as waiting times in queuing models. Because of the (random) linearity of the process, it is easily extended to the modelling of cross-coupled sequences of interarrival and service times. The model can also be extended quite simply to a mixed exponential process with mixed pth order autoregressive and qth order moving average correlation structure, NMEAR (p, q), so that non-Markovian dependence can be handled.

In this paper, we consider two queueing models. Model I is on a single-server queueing system in which the arrival process follows MAP with representation D = (D0,D1) of order m and service time follows phase-type distribution (β, S) of order n. When a customer enters into service, a generalized Erlang clock is started simultaneously. The clock has k stages. The pth stage parameter is θp for 1 ≤ p ≤ k. If a customer completes the service in between the realizations of stages k1 and k2 (1 < k1 < k2 < k) of the clock, it is a perfect one. On the other hand, if the service gets completed either before the kth1 stage realization or after the kth2 stage realization, it is discarded because of imperfection. We analyse this model using the matrix-geometric method. We obtain the expected service time and expected waiting time of a tagged customer. Additional performance measures are also computed. We construct a revenue function and numerically analyse it. In Model II, a single server queueing system in which all assumptions are the same as in Model I except the assumption on service time, is considered. Up to stage k1 service time follows phase-type distribution (α′ , T′) of order n1 and beyond stage k1, the service time follows phase type distribution (β′ , S′) of order n2. We compare the values of the revenue function of the two models

Equivalencies between open and closed queueing networks with finite buffers

The average delay for the GI/G/1 queue is often approximated as a function of the first two moments of interarrival and service times. For highly irregular arrivals, however, it varies widely among queues with the same first two moments, even in moderately heavy traffic. Empirically, it decreases as the interarrival time third moment increases. For GI/M/1 queues, a heavy-traffic expression for the average delay with this property has been previously obtained. The method, however, sheds little light on why the third moment arises. We analyze the equilibrium idle-period distribution in heavy traffic using real-variable methods. For GI/M/1 queues, we derive the above heavy-traffic result and also obtain conditions under which it is either an upper or lower bound. Our approach provides an intuitive explanation for the result and also strongly suggests that similar results should hold for general service. This is supported by empirical evidence. For any given service distribution, it has been conjectured that the expected delay under pure-batch arrivals, where interarrival times are scaled Bernoulli random variables, is an upper bound on the average delay over all interarrival distributions with the same first two moments. We investigate this conjecture and show, among other things, that pure-batch arrivals have the smallest third moment. We obtain conditions under which this conjecture is true and present a counterexample where it fails. Arrivals that arise as overflows from other queues can be highly irregular. We show that interoverflow distributions in a certain class have decreasing failure rate.