On a bulk arrival bulk service infinite service queue

This paper considers an infinite server queue in continuous time in which arrivals are in batches of variable size X and service is provided in groups of fixed size R. We obtain analytical results for the number of busy servers and waiting customers at arbitrary time points. For the number of busy servers, we obtain a recursive relation for the partial binomial moments both in transient and steady states. Special cases are also discussed     

Analysis of a finite MAP/G/1 queue with group services

The finite capacity queues, GI/PH/1/N and PH/G/1/N, in which customers are served in groups of varying sizes were recently introduced and studied in detail by the author. In this paper we consider a finite capacity queue in which arrivals are governed by a particular Markov renewal process, called a Markovian arrival process (MAP). With general service times and with the same type of service rule, we study this finite capacity queueing model in detail by obtaining explicit expressions for (a) the steady-state queue length densities at arrivals, at departures and at arbitrary time points, (b) the probability distributions of the busy period and the idle period of the server and (c) the Laplace-Stieltjes transform of the stationary waiting time distribution of an admitted customer at points of arrivals. Efficient algorithmic procedures for computing the steady-state queue length densities and other system performance measures when services are of phase type are discussed. An illustrative numerical example is presented.     

A server backup model with Markovian arrivals and phase type services

We consider a finite capacity queueing system with one main server who is supported by a backup server. We assume Markovian arrivals, phase type services, and a threshold-type server backup policy with two pre-determined lower and upper thresholds. A request for a backup server is made whenever the buffer size (number of customers in the queue) hits the upper threshold and the backup server is released from the system when the buffer size drops to the lower threshold or fewer at a service completion of the backup server. The request time for the backup server is assumed to be exponentially distributed. For this queuing model we perform the steady state analysis and derive a number of performance measures. We show that the busy periods of the main and backup servers, the waiting times in the queue and in the system, are of phase type. We develop a cost model to obtain the optimal threshold values and study the impact of fixed and variable costs for the backup server on the optimal server backup decisions. We show that the impact of standard deviations of the interarrival and service time distributions on the server backup decisions is quite different for small and large values of the arrival rates. In addition, the pattern of use of the backup server is very different when the arrivals are positively correlated compared to mutually independent arrivals.     

Queues with impolite customers

An interesting behavior of customers arriving to a queue for service concerns the manner in which they join the queue. The arrival discipline of the customers may be impolite, in the sense that an arriving customer who finds all servers busy may pick a position which is not necessarily at the end of the line. We introduce and discuss in detail such an arrival discipline of sufficient generality which has interesting applications. In particular, we show that the more impolite an arrival discipline is, the bigger is the variance of the waiting time. We also study a special model in more depth to provide simple computational formulas for several performance measures.     

The discrete-time MAP/PH/1 queue with multiple working vacations

We consider a discrete-time single-server queueing model where arrivals are governed by a discrete Markovian arrival process (DMAP), which captures both burstiness and correlation in the interarrival times, and the service times and the vacation duration times are assumed to have a general phase-type distributions. The vacation policy is that of a working vacation policy where the server serves the customers at a lower rate during the vacation period as compared to the rate during the normal busy period. Various performance measures of this queueing system like the stationary queue length distribution, waiting time distribution and the distribution of regular busy period are derived. Through numerical experiments, certain insights are presented based on a comparison of the considered model with an equivalent model with independent arrivals, and the effect of the parameters on the performance measures of this model are analyzed.     

A Tandem Queue with a Gate Mechanism

Inspired by a problem regarding cable access networks, we consider a two station tandem queue with Poisson arrivals. At station 1 we operate a gate mechanism, leading to batch arrivals at station 2. Upon arrival at station 1, customers join a queue in front of a gate. Whenever all customers present at the service area of station 1 have received service, the gate before as well as a gate behind the service facility open. Customers leave the service area and enter station 2 (as a batch), while all customers waiting at the gate in front of station 1 are admitted into the service area. For station 1 we analyse the batch size and the time between two successive gate openings, as well as waiting and sojourn times of individual customers for different service disciplines. For station 2, we investigate waiting times of batch customers, where we allow that service times may depend on the size of the batch and also on the interarrival time. In the analysis we use Wiener–Hopf factorization techniques for Markov modulated random walks.     

Analysis of M/M/S queues with 1-limited service

In this paper, a multiple server queue, in which each server takes a vacation after serving one customer is studied. The arrival process is Poisson, service times are exponentially distributed and the duration of a vacation follows a phase distribution of order 2. Servers returning from vacation immediately take another vacation if no customers are waiting. A matrix geometric method is used to find the steady state joint probability of number of customers in the system and busy servers, and the mean and the second moment of number of customers and mean waiting time for this model. This queuing model can be used for the analysis of different kinds of communication networks, such as multi-slotted networks, multiple token rings, multiple server polling systems and mobile communication systems.     

带有止步和中途退出的成批到达的M~x/M/R/N同步休假排队系统的性能分析

研究了带有止步和中途退出的Mx/M/R/N同步休假排队系统.顾客成批到达.到达的顾客如果看到服务员正在休假或者全忙,他或者以概率b决定进入队列等待服务,或者以概率1-b止步(不进入系统).系统根据一定的原则以概率nk在未止步的k个顾客中选择n个进入系统.在系统中排队等待服务的顾客可能因为等待的不耐烦而在没有接受服务的情况下离开系统(中途退出).系统中一旦没有顾客,R个服务员立即进行同步多重休假.首先,利用马尔科夫过程理论建立了系统稳态概率满足的方程组.其次,在证明了相关矩阵可逆性的基础上,利用矩阵解法求出了系统稳态概率的明显表达式,并得到了系统的平均队长、平均等待队长及顾客的平均损失率等性能指标.     

Mixed gated/exhaustive service in a polling model with priorities

In this paper we consider a single-server polling system with switch-over times. We introduce a new service discipline, mixed gated/exhaustive service, that can be used for queues with two types of customers: high and low priority customers. At the beginning of a visit of the server to such a queue, a gate is set behind all customers. High priority customers receive priority in the sense that they are always served before any low priority customers. But high priority customers have a second advantage over low priority customers. Low priority customers are served according to the gated service discipline, i.e. only customers standing in front of the gate are served during this visit. In contrast, high priority customers arriving during the visit period of the queue are allowed to pass the gate and all low priority customers before the gate. We study the cycle time distribution, the waiting time distributions for each customer type, the joint queue length distribution of all priority classes at all queues at polling epochs, and the steady-state marginal queue length distributions for each customer type. Through numerical examples we illustrate that the mixed gated/exhaustive service discipline can significantly decrease waiting times of high priority jobs. In many cases there is a minimal negative impact on the waiting times of low priority customers but, remarkably, it turns out that in polling systems with larger switch-over times there can be even a positive impact on the waiting times of low priority customers.     

A BMAP/SM/1 queue with service times depending on the arrival process

We study a BMAP/>SM/1 queue with batch Markov arrival process input and semi‐Markov service. Service times may depend on arrival phase states, that is, there are many types of arrivals which have different service time distributions. The service process is a heterogeneous Markov renewal process, and so our model necessarily includes known models. At first, we consider the first passage time from level {κ+1} (the set of the states that the number of customers in the system is κ+1) to level {κ} when a batch arrival occurs at time 0 and then a customer service included in that batch simultaneously starts. The service descipline is considered as a LIFO (Last‐In First‐Out) with preemption. This discipline has the fundamental role for the analysis of the first passage time. Using this first passage time distribution, the busy period length distribution can be obtained. The busy period remains unaltered in any service disciplines if they are work‐conserving. Next, we analyze the stationary workload distribution (the stationary virtual waiting time distribution). The workload as well as the busy period remain unaltered in any service disciplines if they are work‐conserving. Based on this fact, we derive the Laplace–Stieltjes transform for the stationary distribution of the actual waiting time under a FIFO discipline. In addition, we refer to the Laplace–Stieltjes transforms for the distributions of the actual waiting times of the individual types of customers. Using the relationship between the stationary waiting time distribution and the stationary distribution of the number of customers in the system at departure epochs, we derive the generating function for the stationary joint distribution of the numbers of different types of customers at departures. This revised version was published online in June 2006 with corrections to the Cover Date.     

Stationary Distributions of GI/M/c Queue with PH Type Vacations

We study a GI/M/c type queueing system with vacations in which all servers take vacations together when the system becomes empty. These servers keep taking synchronous vacations until they find waiting customers in the system at a vacation completion instant.The vacation time is a phase-type (PH) distributed random variable. Using embedded Markov chain modeling and the matrix geometric solution methods, we obtain explicit expressions for the stationary probability distributions of the queue length at arrivals and the waiting time. To compare the vacation model with the classical GI/M/c queue without vacations, we prove conditional stochastic decomposition properties for the queue length and the waiting time when all servers are busy. Our model is a generalization of several previous studies.     

A queueing model with bonus service for certain customers

A queueing model is introduced in which the management has a policy, because of economic reasons, of not operating the service counter unless a certain number, R + 1, of customers are available during each busy period. Thus, the first R customers who arrive must wait until the service counter is opened. Such a policy may cause the management to provide or render additional services to the first R customers. Assuming Poisson arrivals and that both regular and additional services follow exponential distributions, explicit expressions are derived for the stationary queue length and busy period distributions and their expected values. In the special case where R = 1, an explicit expression is presented for the stationary distribution of the waiting time.     

Multi-server retrial queue with negative customers and disasters

We consider a multi-server retrial queue with waiting places in service area and four types of arrivals, positive customers, disasters and two types of negative customers, one for deleting customers in orbit and the other for deleting customers in service area. The four types of arrivals occur according to a Markovian arrival process with marked transitions (MMAP) which may induce the dependence among the arrival processes of the four types. We derive a necessary and sufficient condition for the system to be positive recurrent by comparing sample paths of auxiliary systems whose stability conditions can be obtained. We use a generalized truncated system that is obtained by modifying the retrial rates for an approximation of stationary queue length distribution and show the convergence of approximation to the original model. An algorithmic solution for the stationary queue length distribution and some numerical results are presented.      

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.     

On a Two-Queue Priority System with Impatience and its Application to a Call Center*

We consider an s-server priority system with a protected and an unprotected queue. The arrival rates at the queues and the service rate may depend on the number n of customers being in service or in the protected queue, but the service rate is assumed to be constant for n > s. As soon as any server is idle, a customer from the protected queue will be served according to the FCFS discipline. However, the customers in the protected queue are impatient. If the offered waiting time exceeds a random maximal waiting time I, then the customer leaves the protected queue after time I. If I is less than a given deterministic time, then he leaves the system, else he will be transferred by the system to the unprotected queue. The service of a customer from the unprotected queue will be started if the protected queue is empty and more than a given number of servers become idle. The model is a generalization of the many-server queue with impatient customers. The global balance conditions seem to have no explicit solution. However, the balance conditions for the density of the stationary state process for the subsystem of customers being in service or in the protected queue can be solved. This yields the stability conditions and the probabilities that precisely n customers are in service or in the protected queue. For obtaining performance measures for the unprotected queue, a system approximation based on fitting impatience intensities is constructed. The results are applied to the performance analysis of a call center with an integrated voice-mail-server.     

Open queueing networks in discrete time-some limit theorems

A finite number of nodes, each with a single server and infinite buffers, is considered in discrete time. The service may be FIFO and the service times are constant. The external arrivals and the routing decision variables form a general stationary sequence. Stability of the system is proved under these assumptions. Extension to multiple servers at a node and general stationary distributions holds. If the external input is i.i.d. and the routing is Markovian then stochastic ordering, continuity of stationary distributions, rates of convergence, a functional CLT and a functional LIL and various other limit theorems for the queue length process are also proved. Generalizations to multiple servers at nodes, customers with priority, multiple customer classes, general service length and Markov modulated external arrival cases are discussed.     

A single server queue with service interruptions

In this paper we characterize the queue-length distribution as well as the waiting time distribution of a single-server queue which is subject to service interruptions. Such queues arise naturally in computer and communication problems in which customers belong to different classes and share a common server under some complicated service discipline. In such queues, the viewpoint of a given class of customers is that the server is not available for providing service some of the time, because it is busy serving customers from a different class. A natural special case of these queues is the class of preemptive priority queues. In this paper, we consider arrivals according the Markovian Arrival Process (MAP) and the server is not available for service at certain times. The service times are assumed to have a general distribution. We provide numerical examples to show that our methods are computationally feasible. This revised version was published online in June 2006 with corrections to the Cover Date.     

Non-stationary waiting times in a closed exponential tandem queue

A closed exponential tandem queue in which every customer has to be served by two servers is considered. Given the numbers of customers lined up at each of the two servers, we derive the probability distributions of the waiting time of any customer until his second service is completed, and of the total busy time of the system.     

Analysis of Markov Multiserver Retrial Queues with Negative Arrivals

Negative arrivals are used as a control mechanism in many telecommunication and computer networks. In the paper we analyze multiserver retrial queues; i.e., any customer finding all servers busy upon arrival must leave the service area and re-apply for service after some random time. The control mechanism is such that, whenever the service facility is full occupied, an exponential timer is activated. If the timer expires and the service facility remains full, then a random batch of customers, which are stored at the retrial pool, are automatically removed. This model extends the existing literature, which only deals with a single server case and individual removals. Two different approaches are considered. For the stable case, the matrix–analytic formalism is used to study the joint distribution of the service facility and the retrial pool. The approximation by more simple infinite retrial model is also proved. In the overloading case we study the transient behaviour of the trajectory of the suitably normalized retrial queue and the long-run behaviour of the number of busy servers. The method of investigation in this case is based on the averaging principle for switching processes.     

The $$MMAP\left[ K \right]/PH\left[ K \right]/1$$ queues with a last-come-first-served preemptive service discipline

This paper studies two queueing systems with a Markov arrival process with marked arrivals and PH-distribution service times for each type of customer. Customers (regardless of their types) are served on a last-come-first-served preemptive resume and repeat basis, respectively. The focus is on the stationary distribution of queue strings in the system and busy periods. Efficient algorithms are developed for computing the stationary distribution of queue strings, the mean numbers of customers served in a busy period, and the mean length of a busy period. Comparison is conducted numerically between performance measures of queueing systems with preemptive resume and preemptive repeat service disciplines. A counter-intuitive observation is that for a class of service time distributions, the repeat discipline performs better than the resume one. This revised version was published online in June 2006 with corrections to the Cover Date.