On bulk-service MAP/PH L,N /1/N G-Queues with repeated attempts

We consider a retrial queue with a finite buffer of size N, with arrivals of ordinary units and of negative units (which cancel one ordinary unit), both assumed to be Markovian arrival processes. The service requirements are of phase type. In addition, a PH^L,N bulk service discipline is assumed. This means that the units are served in groups of size at least L, where 1≤ L≤ N. If at the completion of a service fewer than L units are present at the buffer, the server switches off and waits until the buffer length reaches the threshold L. Then it switches on and initiates service for such a group of units. On the contrary, if at the completion of a service L or more units are present at the buffer, all units enter service as a group. Units arriving when the buffer is full are not lost, but they join a group of unsatisfied units called “orbit”. Our interest is in the continuous-time Markov chain describing the state of the queue at arbitrary times, which constitutes a level dependent quasi-birth-and-death process. We start by analyzing a simplified version of our queueing model, which is amenable to numerical calculation and is based on spatially homogeneous quasi-birth-and-death processes. This leads to modified matrix-geometric formulas that reveal the basic qualitative properties of our algorithmic approach for computing performance measures. AMS Subject Classification: Primary 60K25 Secondary 68M20 90B22.     

Analysis of the MAP/G a ,b /1/N Queue

The Markovian arrival process (MAP) is used to represent the bursty and correlated traffic arising in modern telecommunication network. In this paper, we consider a single server finite capacity queue with general bulk service rule in which arrivals are governed by MAP and service times are arbitrarily distributed. The distributions of the number of customers in the queue at arbitrary, post-departure and pre-arrival epochs have been obtained using the supplementary variable and the embedded Markov chain techniques. Computational procedure has been given when the service time distribution is of phase type.     

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 Markov-modulated M/G/1 queue II: Busy period and time for buffer overflow

We consider an M/G/1 queue where the arrival and service processes are modulated by a two state Markov chain. We assume that the arrival rate, service time density and the rates at which the Markov chain switches its state, are functions of the total unfinished work (buffer content) in the queue. We compute asymptotic approximations to performance measures such as the mean residual busy period, mean length of a busy period, and the mean time to reach capacity.This research was supported in part by NSF Grants DMS-84-06110, DMS-85-01535 and DMS-86-20267, and grants from the U.S. Israel Binational Science Foundation and the Israel Academy of Sciences.     

The queue length distributions in the finite buffer bulk-service MAP/G/1 queue with multiple vacations

We consider a finite buffer batch service queueing system with multiple vacations wherein the input process is Markovian arrival process (MAP). The server leaves for a vacation as soon as the system empties and is allowed to take repeated (multiple) vacations. The service- and vacation- times are arbitrarily distributed. We obtain the queue length distributions at service completion, vacation termination, departure, arbitrary and pre-arrival epochs. Finally, some performance measures such as loss probability, average queue lengths are discussed. Computational procedure has been given when the service- and vacation- time distributions are of phase type (PH-distribution).     

Chernoff bounds for mean overflow rates

A number of independent traffic streams arrive at a queueing node which provides a finite buffer and a non-idling service at constant rate. Customers which arrive when the buffer is full are dropped and counted as overflows. We present Chernoff type bounds for mean overflow rates in the form of finite-dimensional minimization problems. The results are based on bounds for moment generating functions of buffer and bandwidth usage of the individual streams in an infinite buffer with constant service rate. We calculate these functions for regulated, Poisson and certain on/off sources. The achievable statistical multiplexing gain and the tightness of the bounds are demonstrated by several numerical examples.     

M/C2/c/K queuing model and optimization for a geriatric care center

Queuing systems with finite buffers are reasonable models for many manufacturing, telecommunication, and healthcare systems. Although some approximations exist, the exact analysis of multi‐server and finite‐buffer queues with general service time distribution is unknown. However, the phase‐type assumption for service time is a frequently used approach. Because the Cox distribution, a kind of phase‐type distribution, provides a good representation of data with great variability, it has a vast area of application in modeling service times. The research focus is twofold. First, a theoretical structure of a multi‐server and finite‐buffer queuing system in which the service time is modeled by the two‐phase Cox distribution is studied. It is focused on finding an efficient solution to the stationary probabilities using the matrix‐geometric method. It is shown that the stationary probability vector can be obtained with the matrix‐geometric method by using level‐dependent rate matrices, and the mean queue length is computed. Second, an empirical analysis of the model is presented. The proposed methodology is applied in a case study concerning the geriatric patients. Some numerical calculations and optimizations are performed by using geriatric data. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Fluid queue driven by aGI/GI/1 queue

Consider a model consisting of two phases: the GI/GI/1 queue and a buffer which is fed by a fluid arriving from a single-server queue. The fluid output from the GI/GI/1 queue is of the on/off type with on- and off-periods distributed as successive busy and idle periods in the GI/GI/1 queue. The fluid pours out of the buffer at a constant rate. The steady-state performance of this model is studied. We derive the Laplace-Stieltjes transform of the stationary distribution function of the buffer content in the case of the M/GI/1 queue in the first phase. It is shown that this distribution depends on the form of the service-time distribution. Therefore, the replacement of an M/GI/1 queue by an M/M/1 queue is not correct, in general. Continuity estimates are derived in the cast where the buffer is fed from the GI/GI/1 queue. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow Russia, 1996, Part II.     

Profit maximization in flexible serial queueing networks

We analyze the tradeoff between efficiency and service quality in tandem systems with flexible servers and finite buffers. We reward efficiency by assuming that a revenue is earned each time a job is completed, and penalize poor service quality by incorporating positive holding costs. We study the dynamic assignment of servers to tasks with the objective of maximizing the long-run average profit. For systems of arbitrary size, structured service rates, and linear or nonlinear holding costs, we determine the server assignment policy that maximizes the profit. For systems with two stations, two servers with arbitrary service rates, and linear holding costs, we show that the optimal server assignment policy is of threshold type and determine the value of this threshold as a function of the revenue and holding cost. The threshold can be interpreted as the best possible buffer size, and hence our results prove the equivalence of addressing service quality via a holding cost and via limiting the buffer size. Furthermore, we identify the optimal buffer size when each buffer space comes at a cost. We provide numerical results that suggest that the optimal policy also has a threshold structure for nonlinear holding costs. Finally, for larger systems with arbitrary service rates, we propose effective server assignment heuristics.     

带有止步和状态相依的M/Ej/1/N排队系统的矩阵解法

研究了带有止步和服务率依赖于状态的M/Ej/1/N排队系统.顾客到达系统时,以一定的概率选择进入系统或止步(不进入系统).顾客接受服务的服务率依赖于系统中的顾客数,当系统中的顾客数不超过临界值k时,服务员慢速服务;否则,服务员快速服务.利用分块矩阵的方法,推出了稳态概率向量所满足的矩阵形式的迭代公式,给出了稳态概率的表达式和计算过程.作为特例,考虑了N=4时系统稳态概率的计算.在此基础上,还求出了系统的一些性能指标,并建立了以临界值k为控制变量的费用模型.通过数值分析,求出了使费用函数最小的最优临界值k*,并进一步研究了模型参数对最优临界值和最优费用的影响.     

Distribution of spatial requirements for a MAP/G/1 queue when space and service times are dependent

In this paper, we consider a MAP/G/1 queue in which each customer arrives with a service and a space requirement, which could be dependent. However, the space and service requirements of different customers are assumed to be independent. Each customer occupies its space requirement in a buffer until it has completely received its service, at which time, it relinquishes the space it occupied. We study and solve the problem of finding the steady-state distribution of the total space requirement of all customers present in the system. In the process of doing so, we also generalize the solution of the MAP/G/1 queue and find the time-average joint distribution of the queue-length, the state of the arrival process and the elapsed service time, conditioned on the server being busy. This problem has applications to the design of buffer requirements for a computer or communication system.     

Expulsion and scheduling control for multiclass queues with heterogeneous servers

We consider an M/M/2 system with nonidentical servers and multiple classes of customers. Each customer class has its own reward rate and holding cost. We may assign priorities so that high priority customers may preempt lower priority customers on the servers. We give two models for which the optimal admission and scheduling policy for maximizing expected discounted profit is determined by a threshold structure on the number of customers of each type in the system. Surprisingly, the optimal thresholds do not depend on the specific numerical values of the reward rates and holding costs, making them relatively easy to determine in practice. Our results also hold when there is a finite buffer and when customers have independent random deadlines for service completion.     

Designing a call center with an IVR (Interactive Voice Response)

A call center is a service operation that caters to customer needs via the telephone. Call centers typically consist of agents that serve customers, telephone lines, an Interactive Voice Response (IVR) unit, and a switch that routes calls to agents. In this paper we study a Markovian model for a call center with an IVR. We calculate operational performance measures, such as the probability for a busy signal and the average wait time for an agent. Exact calculations of these measures are cumbersome and they lack insight. We thus approximate the measures in an asymptotic regime known as QED (Quality and Efficiency Driven) or the Halfin–Whitt regime, which accommodates moderate to large call centers. The approximations are both insightful and easy to apply (for up to 1000’s of agents). They yield, as special cases, known and novel approximations for the M/M/N/N (Erlang-B), M/M/S (Erlang-C) and M/M/S/N queue.     

Optimal static pricing for a service facility with holding costs

We study a service facility modelled as a single-server queueing system with Poisson arrivals and limited or unlimited buffer size. In systems with unlimited buffer size, the service times have general distributions, whereas in finite buffered systems service times are exponentially distributed. Arriving customers enter if there is room in the facility and if they are willing to pay the posted price. The same price is charged to all customers at all times (static pricing). The service provider is charged a holding cost proportional to the time that the customers spend in the system. We demonstrate that there is a unique optimal price that maximizes the long-run average profit per unit time. We also investigate how optimal prices vary as system parameters change. Finally, we consider buffer size as an additional decision variable and show that there is an optimal buffer size level that maximizes profit.     

On the BMAP/G/1 G-queues with second optional service and multiple vacations

This paper deals with an BMAP/G/1 G-queues with second optional service and multiple vacations. Arrivals of positive customers and negative customers follow a batch Markovian arrival process (BMAP) and Markovian arrival process (MAP), respectively. After completion of the essential service of a customer, it may go for a second phase of service. The arrival of a negative customer removes the customer being in service. The server leaves for a vacation as soon as the system empties and is allowed to take repeated (multiple) vacations. By using the supplementary variables method and the censoring technique, we obtain the queue length distributions. We obtain the mean of the busy period based on the renewal theory.     

Performance Analysis of the GI/M/1 Queue with Single Working Vacation and Vacations

In this paper, we consider a new class of the GI/M/1 queue with single working vacation and vacations. When the system become empty at the end of each regular service period, the server first enters a working vacation during which the server continues to serve the possible arriving customers with a slower rate, after that, the server may resume to the regular service rate if there are customers left in the system, or enter a vacation during which the server stops the service completely if the system is empty. Using matrix geometric solution method, we derive the stationary distribution of the system size at arrival epochs. The stochastic decompositions of system size and conditional system size given that the server is in the regular service period are also obtained. Moreover, using the method of semi-Markov process (SMP), we gain the stationary distribution of system size at arbitrary epochs. We acquire the waiting time and sojourn time of an arbitrary customer by the first-passage time analysis. Furthermore, we analyze the busy period by the theory of limiting theorem of alternative renewal process. Finally, some numerical results are presented.     

An abstract approach to nonlinear boltzmann-type equations

Summary Using a general existence and uniqueness theory for linear time dependent kinetic equations, for general inhomogeneous multidimensional spatial and velocity domains and partially absorbing boundaries, we obtain local in time solutions of a class of nonlinear Boltzmann type equations. For small initial-boundary data we obtain global in time solutions. The ideal norm on certain ideals in the Banach space ofL _p-functions on phase space is used to measure the ?size? of initial-boundary data and solutions. Kaniel-Shinbrot type upper and lower approximation arguments are applied. The combined length of the time interval of existence when applying the method repeatedly is analyzed as a function of the size of the initial-boundary data. Specific applications to the nonlinear Boltzmann equation itself and to the plane Broadwell model are given. Research conducted under the auspices of C.N.R. (Consiglio Nazionale delle Ricerche), Gruppo Fisica-Matematica, and partially supported by M.P.I. (Ministero della Pubblica Intruzione). Research conducted as a visiting professor supported by C.N.R., Gruppo Fisica-Matematica. Permanente address: Dept. of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A.     

Optimal inventory replenishment policy for a queueing system with finite waiting room capacity

We treat an inventory control problem in a facility that provides a single type of service for customers. Items used in service are supplied by an outside supplier. To incorporate lost sales due to service delay into the inventory control, we model a queueing system with finite waiting room and non-instantaneous replenishment process and examine the impact of finite buffer on replenishment policies. Employing a Markov decision process theory, we characterize the optimal replenishment policy as a monotonic threshold function of reorder point under the discounted cost criterion. We present a simple procedure that jointly finds optimal buffer size and order quantity.     

Optimal buffer size and dynamic rate control for a queueing system with impatient customers in heavy traffic

We address a rate control problem associated with a single server Markovian queueing system with customer abandonment in heavy traffic. The controller can choose a buffer size for the queueing system and also can dynamically control the service rate (equivalently the arrival rate) depending on the current state of the system. An infinite horizon cost minimization problem is considered here. The cost function includes a penalty for each rejected customer, a control cost related to the adjustment of the service rate and a penalty for each abandoning customer. We obtain an explicit optimal strategy for the limiting diffusion control problem (the Brownian control problem or BCP) which consists of a threshold-type optimal rejection process and a feedback-type optimal drift control. This solution is then used to construct an asymptotically optimal control policy, i.e. an optimal buffer size and an optimal service rate for the queueing system in heavy traffic. The properties of generalized regulator maps and weak convergence techniques are employed to prove the asymptotic optimality of this policy. In addition, we identify the parameter regimes where the infinite buffer size is optimal.