Erlang Loss Model Research Articles

User-centric Markov reward model for state-dependent Erlang loss systems

Markov reward models are commonly used in the analysis of systems by integrating a reward rate to each system state. Typically, rewards are defined based on system states and reflect the system’s perspective. From a user’s point of view, it is important to consider the changing system conditions and dynamics while the user consumes a service. The key contributions of this paper are proper definitions for (i) system-centric reward and (ii) user-centric reward of the Erlang loss model M/M/n-0 and M/M(x)/n with state-dependent service rates, as well as (iii) the analysis of the relationships between those metrics. Our key result allows a simple computation of the user-centric rewards. The differences between the system-centric and the user-centric rewards are demonstrated for a real-world cloud gaming use case. To the best of our knowledge, this is the first analysis showing the relationship between user-centric rewards and system-centric rewards. This work gives relevant and important insights in how to integrate the user’s perspective in the analysis of Markov reward models and is a blueprint for the analysis of other services beyond cloud gaming while also considering user engagement.

A queuing model for ventilator capacity management during the COVID-19 pandemic.

We applied a queuing model to inform ventilator capacity planning during the first wave of the COVID-19 epidemic in the province of British Columbia (BC), Canada. The core of our framework is a multi-class Erlang loss model that represents ventilator use by both COVID-19 and non-COVID-19 patients. Input for the model includes COVID-19 case projections, and our analysis incorporates projections with different levels of transmission due to public health measures and social distancing. We incorporated data from the BC Intensive Care Unit Database to calibrate and validate the model. Using discrete event simulation, we projected ventilator access, including when capacity would be reached and how many patients would be unable to access a ventilator. Simulation results were compared with three numerical approximation methods, namely pointwise stationary approximation, modified offered load, and fixed point approximation. Using this comparison, we developed a hybrid optimization approach to efficiently identify required ventilator capacity to meet access targets. Model projections demonstrate that public health measures and social distancing potentially averted up to 50 deaths per day in BC, by ensuring that ventilator capacity was not reached during the first wave of COVID-19. Without these measures, an additional 173 ventilators would have been required to ensure that at least 95% of patients can access a ventilator immediately. Our model enables policy makers to estimate critical care utilization based on epidemic projections with different transmission levels, thereby providing a tool to quantify the interplay between public health measures, necessary critical care resources, and patient access indicators.

Sensitivity of mean-field fluctuations in Erlang loss models with randomized routing

AbstractIn this paper we study a large system of N servers, each with capacity to process at most C simultaneous jobs; an incoming job is routed to a server if it has the lowest occupancy amongst d (out of N) randomly selected servers. A job that is routed to a server with no vacancy is assumed to be blocked and lost. Such randomized policies are referred to JSQ(d) (Join the Shortest Queue out of d) policies. Under the assumption that jobs arrive according to a Poisson process with rate $N\lambda^{(N)}$ where $\lambda^{(N)}=\sigma-\frac{\beta}{\sqrt{N}\,}$ , $\sigma\in\mathbb{R}_+$ and $\beta\in\mathbb{R}$ , we establish functional central limit theorems for the fluctuation process in both the transient and stationary regimes when service time distributions are exponential. In particular, we show that the limit is an Ornstein–Uhlenbeck process whose mean and variance depend on the mean field of the considered model. Using this, we obtain approximations to the blocking probabilities for large N, where we can precisely estimate the accuracy of first-order approximations.

Weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain

AbstractOur aim is to find sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain. First, we study properties of the state indicator function and the state occupation measure of a Markov chain. In particular, we establish weak convergence of the state occupation measure under a scaling of the generator matrix. Then, relying on the connection between the state occupation measure and the Dynkin martingale, we provide sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure. We apply our results to derive diffusion limits for the Markov-modulated Erlang loss model and the regime-switching Cox–Ingersoll–Ross process.

Mathematical Modelling of Inpatient Flow and Bed Allocation in a Nigerian Tertiary Hospital

Unnecessary delay on queues at hospitals and lack of access to care are associated to loss of lives. In this study we analyze the flow of inpatients in order to find out the appropriate number of beds needed to realize a smooth flow at points where congestion existed in a Nigerian university teaching hospital. Basic hospital statistics, like percentage occupancy, bed turnover rate, etc., are computed and we use queuing model to analyze patient flow and allocation of beds and demonstrate that patient flow in the hospital can be modeled using Erlang loss model.

Modeling Yellow and Red Alert Durations for Ambulance Systems

Emergency systems are designed to almost always have enough capacity to respond to emergencies. However, capacity shortage periods do occur and these systems need to recover quickly. In this study, we apply queueing models and study whether it is better for an emergency system to add or to expedite servers, in order to quickly recover from a capacity shortage period. We focus on emergency medical service (EMS) systems and use Erlang loss models to study Red Alerts (when all ambulances are busy) and Yellow Alerts (when the number of available ambulances falls below a threshold). We analyze two loss models: one with Markovian state‐dependent service rates and one with generally and independently distributed service times. We validate the two models against EMS data sets from two cities. Despite the fact that the distribution of ambulance service times is a mixture of lognormal distributions, which is far from being exponential, we find that the loss model with Markovian state‐dependent service rates provides a better representation of empirical Yellow and Red alert statistics. We build on the model with state‐dependent rates and use the theory of absorbing Markov chains to quantify the impact of adding or expediting ambulances, with respect to two performance measures: (i) the duration of alert periods, and (ii) the number of lost calls. This quantification helps EMS staff (dispatchers and supervisors) to make better decisions to avoid, and to recover from, alert periods. For example, staff should not wait until a Red Alert before adding ambulances, which is a common practice, because the expected number of lost calls rapidly increases as the number of available ambulances at the action epoch decreases.

On the design, implementation, and feasibility of hospital admission services: The admission lounge case

An admission lounge is an emerging type of hospital facility that potentially improves the efficiency and efficacy of the perioperative process. We propose a five-step approach for the design, suitability assessment, and optimisation of an admission lounge. The approach uses a case mix optimisation method to select patients for the admission lounge, clinical ward, or for both. Also, it determines the required admission lounge and clinical ward capacities using an Erlang loss model combined with a novel analytical model. The approach is integrated into a decision support system, which helps hospitals to identify the suitability of the admission lounge concept, optimise its configuration, and identify the potential bed reduction in the clinical ward. The decision support system is validated and tested in a case study of a Dutch hospital using their historical data.

Determining Trunk Lines in Call Centers with Nonstationary Arrivals and Lognormal Service Times

Two important resources in a call center are the number of staff and the number of trunk lines required. In this paper, we focus on the decision of the number of trunk lines that a call center should have. The current practice is to use the Erlang B or the <i>M/M/s/0</i> queueing model which assumes Poisson arrivals, exponential service times, <i>s</i> servers and no places in queue, i.e. no customers can wait. In this paper, we improve on the state of practice in determining the required number of trunk lines, by including two realistic features present in call centers. The first realistic feature is to consider nonstationarity of arrivals. The second feature is to consider the lognormal service time distribution instead of the exponential distribution. There is extensive empirical evidence for both features. In order to carry out our computations we use the results of a paper by Massey and Whitt, Operations Research, 44(6), 1996. We have two main findings. Firstly, we find numerically that in our nonstationary Erlang loss model, <i>M_t/G/s/0</i>, an insensitivity result holds. The blocking probability of arrivals at the call center depends only on the mean of the lognormal service time distribution and not on its variance. Our second finding is that current practice is quite robust. In particular, we find the number of trunk lines required using a stationary Poisson approximation. This approximation assumes stationary Poisson arrivals with an appropriately chosen arrival rate and exponential service times. The approximation does quite well in predicting the number of trunk lines required.

Spectral Efficiency Analysis of GSM Networks in South-South Nigeria

In this paper, the technique of multiplicity was used to analyse GSM network capacity in Nigeria. An evaluation of Enhanced stochastic knapsack was used as an approach for resource sharing in multiservice by adopting the Erlang Loss Model in analyzing the SMS capacity. The offered traffic that is Lost Traffic based was used to dimension the system resources. To actualize this work, measurements were conducted in Benin City and Port Harcourt to determine the best signal characterization of the southern part of Nigeria. Based on the measurement data, a model was developed to predict the traffic intensity of the region. Comparisons were carried out on the different types of frequency hopping and the variant DFH based power was applied in improving the spectral efficiency of the network. The results showed that the spectral efficiency increased, as the number of cell per cluster decreased. The optimal value of the number of cell in the cluster caused reduced interference, since the reduced interference could allow the users to achieve higher rates.

Spectral Efficiency Analysis of GSM Networks in South-South Nigeria

In this paper, the technique of multiplicity was used to analyse GSM network capacity in Nigeria. An evaluation of Enhanced stochastic knapsack was used as an approach for resource sharing in multiservice by adopting the Erlang Loss Model in analyzing the SMS capacity. The offered traffic that is Lost Traffic based was used to dimension the system resources. To actualize this work, measurements were conducted in Benin City and Port Harcourt to determine the best signal characterization of the southern part of Nigeria. Based on the measurement data, a model was developed to predict the traffic intensity of the region. Comparisons were carried out on the different types of frequency hopping and the variant DFH based power was applied in improving the spectral efficiency of the network. The results showed that the spectral efficiency increased, as the number of cell per cluster decreased. The optimal value of the number of cell in the cluster caused reduced interference, since the reduced interference could allow the users to achieve higher rates.

ANALYSIS OF MULTI-RESOURCE LOSS SYSTEM WITH STATE-DEPENDENT ARRIVAL AND SERVICE RATES

In this paper, we study a generalization of the classical multi-dimensional Erlang loss model with state-dependent arrival and service rates, in which customers at arrival occupy random quantities of various resources and release them at departure. Total amount of resources allocated to customers cannot exceed predefined maximum levels. There can be two types of customers: positive customers, which occupy positive quantities of resources, and negative customers, which occupy negative quantities of resources. Negative customers increase the amount of resources available to positive customers and therefore decrease blocking of positive customers caused by lack of resources.

Performance Modeling in Multi-Service Communications Systems with Preemptive Scheduling

This paper presents two separate Markov models to investigate blocking probabilities in multi-service communications systems, when preemptive scheduling is adopted to implement service differentiation. One is the generalized model, which is built as a multi-dimensional Markov chain, based on a variant of the multi-dimensional Erlang loss model. The other is the hierarchical model, which is constructed as a multi-level Markov chain, based on a combination of one-dimensional Erlang loss models. A detailed comparison when applying these two models to a general R- service communications system is presented. This validates the major advantages of the proposed hierarchical model: the closed form expressions of blocking probabilities; the dramatically reduced computational complexity; and the excellent scalability for analyzing larger systems. Furthermore, the analytical values are compared with simulation results for two- and three-service systems. Results show that the proposed hierarchical model provides a high degree of accuracy in the blocking probabilities under different scenarios, especially when the relative arrival rates of the lower priority classes are high.

An efficient modeling and dimensioning approach for integrated streaming and elastic traffic

Recent studies show that both data traffic and real-time traffic grow very fast in wired and wireless networks. To provide better performance guarantee, these applications need efficient network modeling and planning. In this paper, the problem where the total bandwidth of a link is shared by streaming traffic (real time traffic such as voice or video etc.) and elastic traffic (such as data) is studied. Integrating streaming traffic and elastic traffic presents a unique dimensioning problem. This paper considers dimensioning a link to satisfy both quality of service (QoS) requirements for streaming traffic, such as loss probability, and elastic traffic, such as mean waiting (delay) time. The Erlang loss model is applied to streaming traffic and a bursty traffic model is applied to the elastic traffic. Efficient dimensioning algorithms based on classical Markovian models and time-scale decomposition are then proposed. Numerical results show that the proposed methods have good accuracy.

Exact Blocking Time Statistics for the Erlang Loss Model

The Erlang loss model is one of the fundamental tools in queueing theory with many applications to communications networks. For example, in a simple cellular voice network, the Erlang-B blocking formula is the traditional approach to model the proportion of time cellular base-stations are fully loaded in the busy hour. Such steady state results do not provide information on such important questions as: how likely is it that the blocking time in the busy hour exceeds some threshold. Hence we look in detail at the blocking time in the busy hour, or any finite period. We derive the exact distribution function and density as well as the moments and moment generating function of the blocking time, denoted X. In addition we derive the probabilities of zero blocking, P(X=0), and complete blocking, P(X=1 hour), in the busy hour.

Multiserver loss queueing system Polya/G/n/0 with peaked input flow

In this paper, we have proposed to use Polya and Pareto distribution to describe peaked arrival processes in telecommunication networks. We have started the work with traffic measurements and analyses. After seeing that the mean, variance, and probability density function of inter–arrival time and packet length change significantly we decided to apply Polya distribution in the system. With this arrival processes, it is possible to analyse the real loss systems in telecommunication and information networks where the access to the multiple communication channels and servers is dynamic. We study a version of the 'classical' Erlang's loss model M/G/n/0 called Polya/G/n/0. This is a full accessible loss system with Polya input flow (negative binomial distributed number of arrivals in a fixed time interval), general distributed service time and 'n' servers. This model is evaluated by simulation with Pareto distributed inter–arrival time. An algorithm for calculation the state probabilities and blocking is presented. It is shown that the variance of the input stream changes significantly the characteristics of the loss systems.

Generalized Queueing Model for Call Blocking Probability and Resource Utilization in OFDM Wireless Networks

Chen and Chen have dealt with a queueing model for obtaining some performance measures of OFDM wireless systems. Recently, Pla et al. have proposed the multi-class Erlang loss model as a more adequate one. However, the above two models considered only subcarriers as system resources, even though power is also another limited resource of OFDM systems. This letter proposes a generalized queueing model incorporating two kinds of resources, power and subcarriers.

On the nonstationary Erlang loss model

Nonstationary loss queueing system (Erlang model) is considered. We study weak ergodicity, bounds on the rate of convergence, approximations, bounds for limit characteristics.

Dimensioning hospital wards using the Erlang loss model

How many beds must be allocated to a specific clinical ward to meet production targets? When budgets get tight, what are the effects of downsizing a nursing unit? These questions are often discussed by medical professionals, hospital consultants, and managers. In these discussions the occupancy rate is of great importance and often used as an input parameter. Most hospitals use the same target occupancy rate for all wards, often 85%. Sometimes an exception is made for critical care and intensive care units. In this paper we demonstrate that this equity assumption is unrealistic and that it might result in an excessive number of refused admissions, particularly for smaller units. Queuing theory is used to quantify this impact. We developed a decision support system, based on the Erlang loss model, which can be used to evaluate the current size of nursing units. We validated this model with hospital data over the years 2004–2006. Finally, we demonstrate the efficiency of merging departments.

Improved approximations for the Erlang loss model

Stochastic loss networks are often very effective models for studying the random dynamics of systems requiring simultaneous resource possession. Given a stochastic network and a multi-class customer workload, the classical Erlang model renders the stationary probability that a customer will be lost due to insufficient capacity for at least one required resource type. Recently a novel family of slice methods has been proposed by Jung et al. (Proceedings of ACM SIGMETRICS conference on measurement and modeling of computer systems, pp. 407---418, 2008) to approximate the stationary loss probabilities in the Erlang model, and has been shown to provide better performance than the classical Erlang fixed point approximation in many regimes of interest. In this paper, we propose some new methods for loss probability calculation. We propose a refinement of the 3-point slice method of Jung et al. (Proceedings of ACM SIGMETRICS conference on measurement and modeling of computer systems, pp. 407---418, 2008) which exhibits improved accuracy, especially when heavily loaded networks are considered, at comparable computational cost. Next we exploit the structure of the stationary distribution to propose randomized algorithms to approximate both the stationary distribution and the loss probabilities. Whereas our refined slice method is exact in a certain scaling regime and is therefore ideally suited to the asymptotic analysis of large networks, the latter algorithms borrow from volume computation methods for convex polytopes to provide approximations for the unscaled network with error bounds as a function of the computational costs.

Asymptotic Inversion of the Erlang B Formula

The Erlang B formula represents the steady-state blocking probability in the Erlang loss model or $M/M/s/s$ queue. We derive asymptotic expansions for the offered load that matches, for a given number of servers, a certain blocking probability. In addressing this inversion problem we make use of various asymptotic expansions for the incomplete gamma function. A similar inversion problem is investigated for the Erlang C formula.