The single server queue in discrete time‐numerical analysis III

We consider a single server queue in discrete time, in which customers must be served before some limit sojourn time of geometrical distribution. A customer who is not served before this limit leaves the sys- tem. The fact of serving customers, holding them in queue or losing them induce costs. The purpose is to decide when to serve the customers so as to minimize these costs. We use a Markov Decision Process with infinite horizon and discounted criterion. We establish the structural properties of the stochastic dynamic programming operator, and we deduce that the optimal policy is of threshold type, and we compute the threshold explicitly.

We consider networks of queues in discrete time, where the steady state distribution can be computed explicitly in closed form (product form networks): (i) Closed cycles and open tandems of single server FCFS Bernoulli nodes with state dependent service probabilities, where customers flow linearly, (ii) networks of doubly stochastic and geometrical queues (which are discrete time analogues of Kelly’s symmetric, resp. general, servers), where customers of different types move through the network governed by a general routing mechanism and request for service according to general, resp. geometrical, distributions, (iii) networks with batch movements of customers and batch service, where the service and routing mechanism is defined via an abstract transition scheme.

In this paper, we study the total workload process and waiting times in a queueing system with multiple types of customers and a first-come-first-served service discipline. An M/G/1 type Markov chain, which is closely related to the total workload in the queueing system, is constructed. A method is developed for computing the steady state distribution of that Markov chain. Using that steady state distribution, the distributions of total workload, batch waiting times, and waiting times of individual types of customers are obtained. Compared to the GI/M/1 and QBD approaches for waiting times and sojourn times in discrete time queues, the dimension of the matrix blocks involved in the M/G/1 approach can be significantly smaller.

In this contribution we consider an M/M/1 queueing model with general server vacations. Transient and steady state analysis are carried out in discrete time by combinatorial methods. Using weak convergence of discrete-parameter Markov chains we also obtain formulas for the corresponding continuous-time queueing model. As a special case we discuss briefly a queueing system with a T-policy operating.

Errata to Alfa, A. S. 1982. Time-inhomogenous bulk server queue in discrete time: A transportation type problem. Opns. Res. 30 650–658.

This paper studies the queue length distributions of a discrete time bulk server queue with time-inhomogenous compound Poisson input in which service is provided in time-inhomogenous batches and the interarrival times of the server have a general distribution. We assume that the service epochs form a renewal process. This sort of queue occurs during peak periods at intermediate bus stops, where passengers are waiting to catch a bus. In the queueing literature, this problem is known as a transportation type problem. The joint distribution of the number of customers waiting at any epoch and the remaining time before the next arrival of the server is obtained. The results are then extended to cover the case in which the waiting space is limited.

Discrete time queues are the significant tools which are used to analyze and model any kind of digital networks. Discrete time analysis of queues are the most important way to efficiently analyze the today's communication systems based on the digital technology. In this paper the buffer of communication network is analyzed through infinite capacity discrete time queue with multiserver. A model of network buffer is presented and performance measures of the network buffer are evaluated such as Mean number in the queue.

Time-dependent analysis of virtual waiting time behaviour in discrete time queues

For several specific queueing models with a vacation policy, the stationary system occupancy at the beginning of a random slot is distributed as the sum of two independent random variables. One of these variables is the stationary number of customers in an equivalent queueing system with no vacations. For models in continuous time with Poissonian arrivals, this result is well-known, and referred to as stochastic decomposition, with proof provided by Fuhrmann and Cooper. For models in discrete time, this result received less attention, with no proof available to date. In this paper, we first establish a proof of the decomposition result in discrete time. When compared to the proof in continuous time, conditions for the proof in discrete time are somewhat more general. Second, we explore four different examples: non-preemptive priority systems, slot-bound priority systems, polling systems, and fiber delay line (FDL) buffer systems. The first two examples are known results from literature that are given here as an illustration. The third is a new example, and the last one (FDL buffer systems) shows new results. It is shown that in some cases the queueing analysis can be considerably simplified using this decomposition property.

We develop a strongly efficient rare-event simulation algorithm for computing the tail of the steady-state waiting time in a single server queue with regularly varying service times. Our algorithm is based on a state-dependent importance sampling strategy that is constructed so as to be straightforward to implement. The construction of the algorithm and its asymptotic optimality rely on a Lyapunov-type inequality that is used to bound the second moment of the estimator. The solution to the Lyapunov inequality is constructed using fluid heuristics. Our approach takes advantage of the regenerative ratio formula for the steady-state distribution--and does not use the first passage time representation that is particular to the delay in the G/G/1 queue. Hence, the strategy has the potential to be applied in more general queueing models.

[1] We used the frameworks of queuing theory and connectivity to study the runoff generated under constant rainfall on a one-dimensional slope with randomly distributed infiltrability. The equivalence between the stationary runoff-runon equation and the customers waiting time in a single server queue provides a theoretical link between the statistical description of infiltrability and that of runoff flow rate. Five distributions of infiltrability, representing soil heterogeneities at different scales, are considered: four uncorrelated (exponential, bimodal, lognormal, uniform) and one autocorrelated (lognormal, with or without a nugget). The existing theoretical results are adapted to the hydrological framework for the exponential case, and new theoretical developments are proposed for the bimodal law. Numerical simulations validate these results and improve our understanding of runoff-runon for all of the distributions. The quantities describing runoff generation (runoff one-point statistics) and its organization into patterns (patterns statistics and connectivity) are studied as functions of rainfall rate. The variables describing the wet areas are also compared to those describing the rainfall excess areas, i.e., the areas where rainfall exceeds infiltrability. Preliminary results concerning the structural and functional connectivity functions are provided, as well as a discussion about the origin of scale effects in such a system. We suggest that the upslope no-flow boundary condition may be responsible for the dependence of the runoff coefficient on the scale of observation. Queuing theory appears to be a promising framework for runoff-runon modeling and hydrological connectivity problems.

An exact matrix representation of the operator associated with a correlated arrival process is obtained from which an accurate numerical procedure is derived for the computation of the dominant eigenvalue . The peakedness function is studied in order to obtain a useful approximation to and to permit the numerical computation of peakedness. This is important in practice because direct observation of data allows peakedness determination but does not lead directly to the eigenvalue; however, performance measures of queueing models depend directly on Large deviation limit formulae are derived for this correlated arrival process, namely, for the time to the l'th arrival and for the number of arrivals in , which are then used to obtain the decrements (critical exponent coefficients) for the waiting time in a single server queue and for the number in system. A consequence of these formulae is a simple relation between the two decrements. Also, approximations for the distributions of complementary waiting time and number in system are constructed

A generalized SBBP/G/1 queue and its applications

The single server queue is one of the most basic queueing systems to model stochastic waiting dynamics. Most work involving the single server queue only analyzes the customer or agent behavior. However, in this work, we are inspired by COVID-19 applications and are interested in the interaction between customers and more specifically, the time that adjacent customers overlap in the queue. To this end, we derive a new recursion for this overlap time and study the steady state behavior of the overlap time via simulation and probabilistic analysis. We find that the overlap time between adjacent customers in the M/M/1 queue has a conditional distribution that is given by an exponential distribution, however, as the distance between customers grows, the probability of a non-negative overlap time decreases geometrically. We also find via simulation that the exponential distribution still holds when the distribution is non-exponential, hinting at a more general result.

We consider single-server queue, which models a telecommunication node. Packets arrive in the system according to the Markovian arrival process (MAP). The packets bring some amount of information, which is stored in a continuous-type unlimited buffer before it will be transmitted out of the system. We consider the system with fluid manner of service, which means that the server takes the information from the buffer with constant speed. Our aim is to derive the probability distribution of the real waiting time in the system.