Waiting times in a two-queue model with exhaustive and Bernoulli service

This paper deals with waiting times in a two-queue polling system in which one queue is served according to the Bernoulli service discipline and the other one attains exhaustive service. Exact results are derived for the LST's of the waiting time distributions via an iteration scheme. Based on those results the mean waiting times are expressed in the system parameters.     

Monotonicity properties of user equilibrium policies for parallel batch systems

We study a simple network with two parallel batch-service queues, where service at a queue commences when the batch is full and each queue is served by infinitely many servers. A stream of general arrivals observe the current state of the system on arrival and choose which queue to join to minimize their own expected transit time. We show that for each set of parameter values there exists a unique user equilibrium policy and that it possesses various monotonicity properties. User equilibrium policies for probabilistic routing are also discussed and compared with the state-dependent setting.     

Optimal control of a queueing system with an exponential and an Erlangian server and renewal input stream

Customers arrive in a renewal process at a queue which is served by an exponential and a two-stage Erlangian server. We prove the optimal policy for assignment of customers to the servers which for any t maximizes the expected number of served customers in [0,t].     

Dynamic server assignment in a two-queue model

We consider a polling model of two M/G/1 queues, served by a single server. The service policy for this polling model is of threshold type. Service at queue 1 is exhaustive. Service at queue 2 is exhaustive unless the size of queue 1 reaches some level T during a service at queue 2; in the latter case the server switches to queue 1 at the end of that service. Both zero- and nonzero switchover times are considered. We derive exact expressions for the joint queue length distribution at customer departure epochs, and for the steady-state queue-length and sojourn time distributions. In addition, we supply a simple and very accurate approximation for the mean queue lengths, which is suitable for optimization purposes.     

Stationary distributions in a queueing system with vacation times and limited service

This paper deals with a modified M/G/1 queueing system with finite capacity and a walking server. Units waiting are served up to a limited number before the server takes a vacation time and later returns to the queue again. A computational method for the stationary queue length distribution is developed and illustrated with a numerical example. The model was motivated by similar channel access mechanisms in token-ring local area networks.     

Sojourn time distributions for the M/M/1 queue in a Markovian environment

In this paper, we study a single server queue in which both the arrival rate and service rate depend on the state of an external Markov process (called the environment) with a finite state space. Given that the environment is in state j, the mean arrival and service rates are λ_j and μ_j respectively. For such a queue, the queue length distribution is known to be matrix geometric. In this paper, we characterize the Laplace-Stieltjes transform of the sojourn time distribution under four disciplines - last come first served preemptive resume, last come first served, processor sharing and round robin. We also discuss a potential application of this queue in the are of data communication.     

A Communication Multiplexer Problem: Two Alternating Queues with Dependent Randomly-Timed Gated Regime

Two random traffic streams are competing for the service time of a single server (multiplexer). The streams form two queues, primary (queue 1) and secondary (queue 0). The primary queue is served exhaustively, after which the server switches over to queue 0. The duration of time the server resides in the secondary queue is determined by the dynamic evolution in queue 1. If there is an arrival to queue 1 while the server is still working in queue 0, the latter is immediately gated, and the server completes service there only to the gated jobs, upon which it switches back to the primary queue. We formulate this system as a two-queue polling model with a single alternating server and with randomly-timed gated (RTG) service discipline in queue 0, where the timer there depends on the arrival stream to the primary queue. We derive Laplace–Stieltjes transforms and generating functions for various key variables and calculate numerous performance measures such as mean queue sizes at polling instants and at an arbitrary moment, mean busy period duration and mean cycle time length, expected number of messages transmitted during a busy period and mean waiting times. Finally, we present graphs of numerical results comparing the mean waiting times in the two queues as functions of the relative loads, showing the effect of the RTG regime.     

Packet Models Revisited: Tandem and Priority Systems

We examine two extensions of traditional single-node packet-scale queueing models: tandem networks and (strict) priority systems. Two generic input processes are considered: periodic and Poisson arrivals. For the two-node tandem, an exact expression is derived for the joint distribution of the total queue length, and the queue length of the first queue, implicitly determining the distribution of the second queue. Similarly we derive the distribution of the low-priority queue in a two-class priority system. We also provide explicit approximations based on the Brownian bridge.     

On the M/G/1 queue with feedback and optional server vacations based on a single vacation policy

We study a single server queue with batch arrivals and general (arbitrary) service time distribution. The server provides service to customers, one by one, on a first come, first served basis. Just after completion of his service, a customer may leave the system or may opt to repeat his service, in which case this customer rejoins the queue. Further, just after completion of a customer's service the server may take a vacation of random length or may opt to continue staying in the system to serve the next customer. We obtain steady state results in explicit and closed form in terms of the probability generating functions for the number of customers in the queue, the average number of customers and the average waiting time in the queue. Some special cases of interest are discussed and some known results have been derived. A numerical illustration is provided.     

Limited space heterogeneous queueing problem with an additional special feeding source

This paper considers a limited queue space, single channel, first come first served heterogeneous queueing system. There are two operating states sayE and F. For either state, the time in transition from one state to the other, follows an exponential distribution. In both the states E and F, the arrivals are Poissonian with different parameters and the service times have different exponential distributions. In either state, an additional number M of units join the system whenever the number already there declines to one unit. Thus the channel is busy at all times.     

多服务台并联排队系统的随机模拟

针对多服务台并联排队系统,提出了计算机随机模拟的方法.分别研究了基本和非基本排队系统中,诸如排队人数、排队时间、平稳分布、忙期及其忙期服务人数等各项排队指标的模拟求解方法.通过模拟例子和实际数据分析,显示了本文所提模拟方法的有效性、灵活性和实用性.     

Single-Server Queues with Markov-Modulated Arrivals and Service Speed

This paper considers single-server queues with several customer classes. Arrivals of customers are governed by the underlying continuous-time Markov chain with finite states. The distribution of the amount of work brought into the system on arrival is assumed to be general, which may differ with different classes. Further, the service speed depends on the state of the underlying Markov chain. We first show that given such a queue, we can construct the corresponding new queue with constant service speed by means of a change of time scale, and the time-average quantities of interest in the original queue are given in terms of those in the new queue. Next we characterize the joint distribution of the length of a busy period and the number of customers served during the busy period in the original queue. Finally, assuming the FIFO service discipline, we derive the Laplace–Stieltjes transform of the actual waiting time distribution in the original queue.     

A finite capacity multi-queueing system with priorities and with repeated server vacations

In this paper, we study an M/G/1 multi-queueing system consisting ofM finite capacity queues, at which customers arrive according to independent Poisson processes. The customers require service times according to a queue-dependent general distribution. Each queue has a different priority. The queues are attended by a single server according to their priority and are served in a non-preemptive way. If there are no customers present, the server takes repeated vacations. The length of each vacation is a random variable with a general distribution function. We derive steady state formulas for the queue length distribution and the Laplace transform of the queueing time distribution for each queue.     

A single-server batch arrival queue with returning customers

We consider a new class of batch arrival retrial queues. By contrast to standard batch arrival retrial queues we assume if a batch of primary customers arrives into the system and the server is free then one of the customers starts to be served and the others join the queue and then are served according to some discipline. With the help of Lyapunov functions we have obtained a necessary and sufficient condition for ergodicity of embedded Markov chain and the joint distribution of the number of customers in the queue and the number of customers in the orbit in steady state. We also have suggested an approximate method of analysis based on the corresponding model with losses.     

Effective bandwidths at multi-class queues

Consider a queue which serves traffic from a number of distinct sources and which is required to deliver a performance guarantee, expressed in terms of the mean delay or the probability the delay exceeds a threshold. For various simple models we show that an effective bandwidth can be associated with each source, and that the queue can deliver its performance guarantee by limiting the sources served so that their effective bandwidths sum to less than the capacity of the queue.     

Two competing discrete-time queues with priority

This paper considers a class of two discrete-time queues with infinite buffers that compete for a single server. Tasks requiring a deterministic amount of service time, arrive randomly to the queues and have to be served by the server. One of the queues has priority over the other in the sense that it always attempts to get the server, while the other queue attempts only randomly according to a rule that depends on how long the task at the head of the queue has been waiting in that position. The class considered is characterized by the fact that if both queues compete and attempt to get the server simultaneously, then they both fail and the server remains idle for a deterministic amount of time. For this class we derive the steady-state joint generating function of the state probabilities. The queueing system considered exhibits interesting behavior, as we demonstrate by an example.     

Exact Analysis of the State-Dependent Polling Model

We consider a polling model in which a number of queues are served, in cyclic order, by a single server. Each queue has its own distinct Poisson arrival stream, service time, and switchover time (the server's travel time from that queue to the next) distribution. A setup time is incurred if the polled queue has one or more customers present. This is the polling model with State-Dependent service (the SD model). The SD model is inherently complex; hence, it has often been approximated by the much simpler model with State-Independent service (the SI model) in which the server always sets up for a service at the polled queue, regardless of whether it has customers or not. We provide an exact analysis of the SD model and obtain the probability generating function of the joint queue length distribution at a polling epoch, from which the moments of the waiting times at the various queues are obtained. A number of numerical examples are presented, to reveal conditions under which the SD model could perform worse than the corresponding SI model or, alternately, conditions under which the SD model performs better than a corresponding model in which all setup times are zero. We also present expressions for a variant of the SD model, namely, the SD model with a patient server.     

Revenue-maximizing pricing and scheduling strategies in service systems with flexible customers

We propose a model of service operations systems in which customers are heterogeneous both in terms of their private delay sensitivity and flexibility. A service provider maximizes revenue through jointly optimal pricing and steady-state scheduling strategies. We provide a complete analysis for this generally intractable problem. Interestingly, when one queue accommodates a large population of impatient customers, it may be desirable to strategically idle the server in the other queue, which is a phenomenon new to the literature.     

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.