Networks of infinite-server queues with nonstationary Poisson input

In this paper we focus on networks of infinite-server queues with nonhomogeneous Poisson arrival processes. We start by introducing a more general Poisson-arrival-location model (PALM) in which arrivals move independently through a general state space according to a location stochastic process after arriving according to a nonhomogeneous Poisson process. The usual open network of infinite-server queues, which is also known as a linear population process or a linear stochastic compartmental model, arises in the special case of a finite state space. The mathematical foundation is a Poisson-random-measure representation, which can be obtained by stochastic integration. It implies a time-dependent product-form result: For appropriate initial conditions, the queue lengths (numbers of customers in disjoint subsets of the state space) at any time are independent Poisson random variables. Even though there is no dependence among the queue lengths at each time, there is important dependence among the queue lengths at different times. We show that the joint distribution is multivariate Poisson, and calculate the covariances. A unified framework for constructing stochastic processes of interest is provided by stochastically integrating various functionals of the location process with respect to the Poisson arrival process. We use this approach to study the flows in the queueing network; e.g., we show that the aggregate arrival and departure processes at a given queue (to and from other queues as well as outside the network) are generalized Poisson processes (without necessarily having a rate or unit jumps) if and only if no customer can visit that queue more than once. We also characterize the aggregate arrival and departure processes when customers can visit the queues more frequently. In addition to obtaining structural results, we use the stochastic integrals to obtain explicit expressions for time-dependent means and covariances. We do this in two ways. First, we decompose the entire network into a superposition of independent networks with fixed deterministic routes. Second, we make Markov assumptions, initially for the evolution of the routes and finally for the entire location process. For Markov routing among the queues, the aggregate arrival rates are obtained as the solution to a system of input equations, which have a unique solution under appropriate qualifications, but not in general. Linear ordinary differential equations characterize the time-dependent means and covariances in the totally Markovian case.     

A note on stable flow-equivalent aggregation in closed networks

We introduce the Conditional Mean Value Analysis (CMVA) algorithm, an exact solution method for product-form load-dependent closed queueing networks that provides a numerically stable solution of models where the load-dependent Mean Value Analysis (MVA) is numerically unstable. Similarly to the MVA algorithm for constant-rate queues, CMVA performs operations in terms of mean quantities only, i.e., queue-lengths, throughput, response times. Numerical stability derives from a new version of the MVA arrival theorem for load-dependent models which is expressed in terms of mean queue-lengths instead of marginal probabilities. The formula is obtained by the analysis of the conditional state spaces which describe network equilibrium as seen by jobs during their residence times at queues. We also provide a generalization of CMVA to multiclass models that preserves the numerical stability property.     

Two-parameter heavy-traffic limits for infinite-server queues with dependent service times

This paper is a sequel to our 2010 paper in this journal in which we established heavy-traffic limits for two-parameter processes in infinite-server queues with an arrival process that satisfies an FCLT and i.i.d. service times with a general distribution. The arrival process can have a time-varying arrival rate. In particular, an FWLLN and an FCLT were established for the two-parameter process describing the number of customers in the system at time t that have been so for a duration y. The present paper extends the previous results to cover the case in which the successive service times are weakly dependent. The deterministic fluid limit obtained from the new FWLLN is unaffected by the dependence, whereas the Gaussian process limit (random field) obtained from the FCLT has a term resulting from the dependence. Explicit expressions are derived for the time-dependent means, variances, and covariances for the common case in which the limit process for the arrival process is a (possibly time scaled) Brownian motion.     

The 1-Center and 1-Highway problem revisited

We consider a two-queue polling model in which customers upon arrival join the shorter of two queues. Customers arrive according to a Poisson process and the service times in both queues are independent and identically distributed random variables having the exponential distribution. The two-dimensional process of the numbers of customers at the queue where the server is and at the other queue is a two-dimensional Markov process. We derive its equilibrium distribution using two methodologies: the compensation approach and a reduction to a boundary value problem.     

Analyzing <Emphasis Type="Italic">M/M/</Emphasis>1 Queues with Perturbations in the Arrival Process

This paper investigates when the M/M/1 model can be used to predict accurately the operating characteristics of queues with arrival processes that are slightly different from the Poisson process assumed in the model. The arrival processes considered here are perturbed Poisson processes. The perturbations are deviations from the exponential distribution of the inter-arrival times or from the assumption of independence between successive inter-arrival times. An estimate is derived for the difference between the expected numbers in perturbed and M/M/1 queueing systems with the same traffic intensity. The results, for example, indicate that the M/M/1 model can predict the performance of the queue when the arrival process is perturbed by inserting a few short inter-arrival times, an occasional batch arrival or small dependencies between successive inter-arrival times. In contrast, the M/M/1 is not a good model when the arrival process is perturbed by inserting a few long inter-arrival times.     

Stability,monotonicity and invariant quantities in general polling systems

Consider a polling system withK1 queues and a single server that visits the queues in a cyclic order. The polling discipline in each queue is of general gated-type or exhaustive-type. We assume that in each queue the arrival times form a Poisson process, and that the service times, the walking times, as well as the set-up times form sequences of independent and identically distributed random variables. For such a system, we provide a sufficient condition under which the vector of queue lengths is stable. We treat several criteria for stability: the ergodicity of the process, the geometric ergodicity, and the geometric rate of convergence of the first moment. The ergodicity implies the weak convergence of station times, intervisit times and cycle times. Next, we show that the queue lengths, station times, intervisit times and cycle times are stochastically increasing in arrival rates, in service times, in walking times and in setup times. The stability conditions and the stochastic monotonicity results are extended to the polling systems with additional customer routing between the queues, as well as bulk and correlated arrivals. Finally, we prove that the mean cycle time, the mean intervisit time and the mean station times are invariant under general service disciplines and general stationary arrival and service processes.     

Performance analysis of an ISDN switch with distributed architecture: circuit switched calls

We have developed an approximate analytic model and a detailed simulation model to study the performance of an ISDN switch with distributed architecture. The analytic model treats the switch as a network of single server and infinite server queues with nonpreemptive priority service, general service times and batch arrivals. The simulation program is written in a distributed and modular way so as to simplify model development and debugging. Also extensive statistical techniques are employed for simulation output validation. It is observed that the analytic and the simulation models are in close agreement for the mean end-to-end delay and in moderately close agreement for the 95th percentile points of the end-to-end delay distribution. The comparisons between the analytic and the simulation models lead us to conjecture that the analytic model would be even more accurate for bigger systems with several hundred processors (where simulation models are too expensive to run). Even though the model assumes Poisson external call arrival process, it is shown that it may be applied with reasonable accuracy even when external call arrivals are non-Poisson. This is due to the fact that the composite message arrival process at a processor or transmission element tends to be close to Poisson even when the external call arrivals are non-Poisson.     

Infinite-server systems with Coxian arrivals

We consider a network of infinite-server queues where the input process is a Cox process of the following form: The arrival rate is a vector-valued linear transform of a multivariate generalized (i.e., being driven by a subordinator rather than a compound Poisson process) shot-noise process. We first derive some distributional properties of the multivariate generalized shot-noise process. Then these are exploited to obtain the joint transform of the numbers of customers, at various time epochs, in a single infinite-server queue fed by the above-mentioned Cox process. We also obtain transforms pertaining to the joint stationary arrival rate and queue length processes (thus facilitating the analysis of the corresponding departure process), as well as their means and covariance structure. Finally, we extend to the setting of a network of infinite-server queues.

     

Distributional Form of Little's Law for FIFO Queues with Multiple Markovian Arrival Streams and Its Application to Queues with Vacations

This paper considers stationary queues with multiple arrival streams governed by an irreducible Markov chain. In a very general setting, we first show an invariance relationship between the time-average joint queue length distribution and the customer-average joint queue length distribution at departures. Based on this invariance relationship, we provide a distributional form of Little's law for FIFO queues with simple arrivals (i.e., the superposed arrival process has the orderliness property). Note that this law relates the time-average joint queue length distribution with the stationary sojourn time distributions of customers from respective arrival streams. As an application of the law, we consider two variants of FIFO queues with vacations, where the service time distribution of customers from each arrival stream is assumed to be general and service time distributions of customers may be different for different arrival streams. For each queue, the stationary waiting time distribution of customers from each arrival stream is first examined, and then applying the Little's law, we obtain an equation which the probability generating function of the joint queue length distribution satisfies. Further, based on this equation, we provide a way to construct a numerically feasible recursion to compute the joint queue length distribution.     

Queues with Service Rate Controlled by a Delayed Feedback

We consider a single queue with a Markov modulated Poisson arrival process. Its service rate is controlled by a scheduler. The scheduler receives the workload information from the queue after a delay. This queue models the buffer in an earth station in a satellite network where the scheduler resides in the satellite. We obtain the conditions for stability, rates of convergence to the stationary distribution and the finiteness of the stationary moments. Next we extend these results to the system where the scheduler schedules the service rate among several competing queues based on delayed information about the workloads in the different queues.     

Probabilistic proof of the interchangeability of ./M/1 queues in series

Given a finite number of empty ./M/1 queues, let customers arrive according to an arbitrary arrival process and be served at each queue exactly once, in some fixed order. The process of departing customers from the network has the same law, whatever the order in which the queues are visited. This remarkable result, due to R. Weber [4], is given a simple probabilistic proof.     

Ergodicity and analysis of the process describing the system state in polling systems with two queues

Consider a polling system of two queues served by a single server that visits the queues in cyclic order. The polling discipline in each queue is of exhaustive-type, and zero-switchover times are considered. We assume that the arrival times in each queue form a Poisson process and that the service times form sequences of independent and identically distributed random variables, except for the service distribution of the first customer who is served at each polling instant (the time in which the server moves from one queue to the other one). The sufficient and necessary conditions for the ergodicity of such polling system are established as well as the stationary distribution for the continuous-time process describing the state of the system. The proofs rely on the combination of three embedded processes that were previously used in the literature. An important result is that ρ=1 can imply ergodicity in one specific case, where ρ is the typical traffic intensity for polling systems, and ρ<1 is the classical non-saturation condition.     

Vacation models in discrete time

A class of single server vacation queues which have single arrivals and non-batch service is considered in discrete time. It is shown that provided the interarrival, service, vacation, and server operational times can be cast with Markov-based representation then this class of vacation model can be studied as a matrix–geometric or a matrix-product problem – both in the matrix–analytic family – thereby allowing us to use well established results from Neuts (1981). Most importantly it is shown that using discrete time approach to study some vacation models is more appropriate and makes the models much more algorithmically tractable. An example is a vacation model in which the server visits the queue for a limited duration. The paper focuses mainly on single arrival and single unit service systems which result in quasi-birth-and-death processes. The results presented in this paper are applicable to all this class of vacation queues provided the interarrival, service, vacation, and operational times can be represented by a finite state Markov chain.An erratum to this article can be found at     

Analysis of job transfer policies in systems with unreliable servers

We consider a system where incoming jobs may be executed at different servers, each of which goes through alternating periods of being available and unavailable. Neither the states of the servers nor the relevant queue sizes are known at moments of arrival. Hence, a load balancing mechanism that relies on random time-out intervals and job transfers from one queue to another is adopted. The object is to minimize a cost function which may include holding costs and transfer costs. A model of a single queue with an unreliable server and timeouts is analyzed first. The results are then used to obtain an approximate solution for arbitrary number of queues. Several transfer policies are evaluated and compared.     

The MMCPP/GE/c Queue

We obtain the queue length probability distribution at equilibrium for a multi-server, single queue with generalised exponential (GE) service time distribution and a Markov modulated compound Poisson arrival process (MMCPP) – i.e., a Poisson point process with bulk arrivals having geometrically distributed batch size whose parameters are modulated by a Markovian arrival phase process. This arrival process has been considered appropriate in ATM networks and the GE service times provide greater flexibility than the more conventionally assumed exponential distribution. The result is exact and is derived, for both infinite and finite capacity queues, using the method of spectral expansion applied to the two dimensional (queue length by phase of the arrival process) Markov process that describes the dynamics of the system. The Laplace transform of the interdeparture time probability density function is then obtained. The analysis therefore could provide the basis of a building block for modelling networks of switching nodes in terms of their internal arrival processes, which may be both correlated and bursty.     

Inter-Departure Times in Base-Stock Inventory-Queues

An inventory-queue is an inventory system controlled by a processing station with queueing. They are natural building blocks for supply chain models. An important and largely open issue for inventory queues is the characterization of their departure processes. In an inventory-queue, departures are triggered either by a job arrival when the output buffer is not empty or otherwise by a service completion. Such departures are more difficult to analyze than departures from a standard queue. The main results in this study are expressions for the probability distributions and squared coefficient of variations of inter-departure times for base-stock inventory-queues with birth–death production processes.     

On the distributions of infinite server queues with batch arrivals

Queues that feature multiple entities arriving simultaneously are among the oldest models in queueing theory, and are often referred to as “batch” (or, in some cases, “bulk”) arrival queueing systems. In this work, we study the effect of batch arrivals on infinite server queues. We assume that the arrival epochs occur according to a Poisson process, with treatment of both stationary and non-stationary arrival rates. We consider both exponentially and generally distributed service durations, and we analyze both fixed and random arrival batch sizes. In addition to deriving the transient mean, variance, and moment-generating function for time-varying arrival rates, we also find that the steady-state distribution of the queue is equivalent to the sum of scaled Poisson random variables with rates proportional to the order statistics of its service distribution. We do so through viewing the batch arrival system as a collection of correlated sub-queues. Furthermore, we investigate the limiting behavior of the process through a batch scaling of the queue and through fluid and diffusion limits of the arrival rate. In the course of our analysis, we make important connections between our model and the harmonic numbers, generalized Hermite distributions, and truncated polylogarithms.

     

Monotonicity and stability of periodic polling models

This paper deals with the stability of periodic polling models with a mixture of service policies. Customers arrive according to independent Poisson processes. The service times and the switchover times are independent with general distributions. The necessary and sufficient condition for the stability of such polling systems is established. The proof is based on the stochastic monotonicity of the state process at the polling instants. The stability of only a subset of the queues is also analyzed and, in case of heavy traffic, the order of explosion of the queues is given. The results are valid for a model with set-up times, and also when there is a local priority rule at the queues.This work was supported in part by a Fellowship of the Netherlands Organization for Scientific Research NWO-ECOZOEK.     

Finite-Buffer Queues with Workload-Dependent Service and Arrival Rates

We consider M/G/1 queues with workload-dependent arrival rate, service speed, and restricted accessibility. The admittance of customers typically depends on the amount of work found upon arrival in addition to its own service requirement. Typical examples are the finite dam, systems with customer impatience and queues regulated by the complete rejection discipline. Our study is motivated by queueing scenarios where the arrival rate and/or speed of the server depends on the amount of work present, like production systems and the Internet.First, we compare the steady-state distribution of the workload in two finite-buffer models, in which the ratio of arrival and service speed is equal. Second, we find an explicit expression for the cycle maximum in an M/G/1 queue with workload-dependent arrival and service rate. And third, we derive a formal solution for the steady-state workload density in case of restricted accessibility. The proportionality relation between some finite and infinite-buffer queues is extended. Level crossings and Volterra integral equations play a key role in our approach.AMS subject classification: 60K25, 90B22     

Asymptotically optimal open-loop load balancing

In many distributed computing systems, stochastically arriving jobs need to be assigned to servers with the objective of minimizing waiting times. Many existing dispatching algorithms are basically included in the SQ(d) framework: Upon arrival of a job, \(d\ge 2\) servers are contacted uniformly at random to retrieve their state and then the job is routed to a server in the best observed state. One practical issue in this type of algorithm is that server states may not be observable, depending on the underlying architecture. In this paper, we investigate the assignment problem in the open-loop setting where no feedback information can flow dynamically from the queues back to the controller, i.e., the queues are unobservable. This is an intractable problem, and unless particular cases are considered, the structure of an optimal policy is not known. Under mild assumptions and in a heavy-traffic many-server limiting regime, our main result proves the optimality of a subset of deterministic and periodic policies within a wide set of (open-loop) policies that can be randomized or deterministic and can be dependent on the arrival process at the controller. The limiting value of the scaled stationary mean waiting time achieved by any policy in our subset provides a simple approximation for the optimal system performance.