On the M/M/c/N Call Center Queue Modeling and Analysis

The M/M/c/c model is the most widely applied queueing model in the mathematical analysis of call centers. The M/M/c/c model is also referred to as the Erlang Loss System. The Erlang loss model does not take into consideration system attributes such as blocking and busy signals, balking and reneging, retrials and returns. Although, the Erlang loss model is analytically tractable, it is not easy to obtain insight from its results. The need to develop a more accurate call center model has necessitated the modification of the Erlang loss model. In this research, we model and analyze a call center using M/M/c/N the model. The goal of this paper is to extend existing results and prove new results with regards to the monotonicity and limiting behaviour of the M/M/c/N model with respect to the system capacity N

A Study of a Loss System with Priorities

The Erlang loss formula, also known as the Erlang B formula, has been known for over a century and has been used in a wide range of applications, from telephony to hospital intensive care unit management. It provides the blocking probability of arriving customers to a loss system involving a finite number of servers without a waiting room. Because of the need to introduce priorities in many services, an extension of the Erlang B formula to the case of a loss system with preemptive priority is valuable and essential. This paper analytically establishes the consistency between the global balance (steady state) equations for a loss system with preemptive priorities and a known result obtained using traffic loss arguments for the same problem. This paper, for the first time, derives this known result directly from the global balance equations based on the relevant multidimensional Markov chain. The paper also addresses the question of whether or not the well-known insensitivity property of the Erlang loss system is also applicable to the case of a loss system with preemptive priorities, provides explanations, and demonstrates through simulations that, except for the blocking probability of the highest priority customers, the blocking probabilities of the other customers are sensitive to the holding time distributions and that a higher variance of the service time distribution leads to a lower blocking probability of the lower priority traffic

Properties of Recurrent Equations for the Full-Availability Group with BPP Traffic

The paper proposes a formal derivation of recurrent equations describing the occupancy distribution in the full-availability group with multirate Binomial-Poisson-Pascal (BPP) traffic. The paper presents an effective algorithm for determining the occupancy distribution on the basis of derived recurrent equations and for the determination of the blocking probability as well as the loss probability of calls of particular classes of traffic offered to the system. A proof of the convergence of the iterative process of estimating the average number of busy traffic sources of particular classes is also given in the paper

Estimating Bed Requirements for a Pediatric Department in a University Hospital in Egypt

Every day, a considerable number of children in need for health monitoring and control are turned away because of lack of beds in the Pediatric department in Zagazig University hospital in Egypt. This paper estimates the required number of beds needed for controlling this number of turned away children. The paper also investigates the effect of redistributing beds among different specialties on the service level. An Erlang Loss model is applied for estimating required capacity, then an optimization model is used for finding the optimum bed distribution that minimize number of turned away children

Efficient estimation of blocking probabilities in non-stationary loss networks

This paper considers estimation of blocking probabilities in a nonstationary loss network. Invoking the so called MOL (Modified Offered Load) approximation, the problem is transformed into one requiring the solution of blocking probabilities in a sequence of stationary loss networks with time varying loads. To estimate the blocking probabilities Monte Carlo simulation is used and to increase the efficiency of the simulation, we develop a likelihood ratio method that enables samples drawn at a one time point to be used at later time points. This reduces the need to draw new samples every time independently as a new time point is considered, thus giving substantial savings in the computational effort of evaluating time dependent blocking probabilities. The accuracy of the method is analyzed by using Taylor series approximations of the variance indicating the direct dependence of the accuracy on the rate of change of the actual load. Finally, three practical applications of the method are provided along with numerical examples to demonstrate the efficiency of the method

Study of Queuing Systems with a Generalized Departure Process

This work was supported by the Bulgarian National Science Fund under grant BY-TH-105/2005.This paper deals with a full accessibility loss system and a single server delay system with a Poisson arrival process and state dependent exponentially distributed service time. We use the generalized service flow with nonlinear state dependence mean service time. The idea is based on the analytical continuation of the Binomial distribution and the classic M/M/n/0 and M/M/1/k system. We apply techniques based on birth and death processes and state-dependent service rates. We consider the system M/M(g)/n/0 and M/M(g)/1/k (in Kendal notation) with a generalized departure process Mg. The output intensity depends nonlinearly on the system state with a defined parameter: “peaked factor p”. We obtain the state probabilities of the system using the general solution of the birth and death processes. The influence of the peaked factor on the state probability distribution, the congestion probability and the mean system time are studied. It is shown that the state-dependent service rates changes significantly the characteristics of the queueing systems. The advantages of simplicity and uniformity in representing both peaked and smooth behaviour make this queue attractive in network analysis and synthesis

STOCHASTIC MODELING AND TIME-TO-EVENT ANALYSIS OF VOIP TRAFFIC

Voice over IP (VoIP) systems are gaining increased popularity due to the cost effectiveness, ease of management, and enhanced features and capabilities. Both enterprises and carriers are deploying VoIP systems to replace their TDM-based legacy voice networks. However, the lack of engineering models for VoIP systems has been realized by many researchers, especially for large-scale networks. The purpose of traffic engineering is to minimize call blocking probability and maximize resource utilization. The current traffic engineering models are inherited from the legacy PSTN world, and these models fall short from capturing the characteristics of new traffic patterns. The objective of this research is to develop a traffic engineering model for modern VoIP networks. We studied the traffic on a large-scale VoIP network and collected several billions of call information. Our analysis shows that the traditional traffic engineering approach based on the Poisson call arrival process and exponential holding time fails to capture the modern telecommunication systems accurately. We developed a new framework for modeling call arrivals as a non-homogeneous Poisson process, and we further enhanced the model by providing a Gaussian approximation for the cases of heavy traffic condition on large-scale networks. In the second phase of the research, we followed a new time-to-event survival analysis approach to model call holding time as a generalized gamma distribution and we introduced a Call Cease Rate function to model the call durations. The modeling and statistical work of the Call Arrival model and the Call Holding Time model is constructed, verified and validated using hundreds of millions of real call information collected from an operational VoIP carrier network. The traffic data is a mixture of residential, business, and wireless traffic. Therefore, our proposed models can be applied to any modern telecommunication system. We also conducted sensitivity analysis of model parameters and performed statistical tests on the robustness of the models’ assumptions. We implemented the models in a new simulation-based traffic engineering system called VoIP Traffic Engineering Simulator (VSIM). Advanced statistical and stochastic techniques were used in building VSIM system. The core of VSIM is a simulation system that consists of two different simulation engines: the NHPP parametric simulation engine and the non-parametric simulation engine. In addition, VSIM provides several subsystems for traffic data collection, processing, statistical modeling, model parameter estimation, graph generation, and traffic prediction. VSIM is capable of extracting traffic data from a live VoIP network, processing and storing the extracted information, and then feeding it into one of the simulation engines which in turn provides resource optimization and quality of service reports