Erlang Formula Research Articles

Development of a mathematical model for assessing the quality of service on a packet switching subnet

Currently, data traffic is growing rapidly, and ensuring optimal network performance and effective data flow management have become the most important tasks. In this context, the quality of network service plays a crucial role in achieving these goals. This article suggests an approach to solving the problem of efficient service in ISDN. Namely, optimization of resource distribution between channel switching and packet switching subnets in ISDN to calculate optimal quality of service characteristics. In the process of ISDN design analysis, an optimization problem is compiled, where the evaluation of the packet-switched subnet service is used as an objective function, and the evaluation of the circuit-switched subnet service is used as one of the constraints for this task. To calculate the main characteristics of a packet-switched subnet, the subnet is considered as a service system with a delay. During the study, the methods of optimal movement of the generalized channel boundary between the subnets of channel switching and packet switching were identified, depending on the data parameters and the state of the integrated network, which made it possible to develop an optimal mathematical model of optimal control of the generalized boundary. To calculate the bandwidth for channel switching and packet switching subnets, an algorithm has been compiled to implement the resulting model and a program in C++ has been compiled. The study of the generalized boundary and the dynamic redistribution of bandwidth between subnets represents a new approach to network optimization. The results are based on the use of the classical Erlang formula for systems with service failures and on load distribution plans, which makes it possible to effectively manage the maintenance process in the network

From Erlang formula to robust control

From Erlang formula to robust control

Оценка времени обслуживания судна терминалом в системе транспортного узла

It is proposed to evaluate various options for the mooring of a vessel at the berth of a cargo terminal by integer time intervals, which, when using Erlang's formulas, generalized to the case of the Markov process, can be used to solve the problems of planning loading and unloading operations on the company's ships, and should be presented in the form of a vector, the size of which is determined by the number of possible states of the "ship – berth" system.
 A method for assessing the time of servicing a loading and unloading operation by the terminal has been compiled, when observing a number of particulars corresponding to the usual histogram, and each value of the time interval corresponds to a certain bit of the histogram, a variant of evaluating the mathematical expectation of the time the terminal serves one loading and unloading operation.
 It was found that using the least squares method, it is possible to identify the value of the variance of the time of the loading and unloading operation, which, in turn, can be used as a point estimate when compiling the confidence interval for the mathematical expectation of this interval.

Analiza performansi meta-organizacija modeliranih kao pozivni centri za podršku dinamičke rekonfigurabilnosti mrežnih i virtualnih organizacija

Various companies choose to outsource the delivery of part of their services, so as not to deviate from its core business and improve the service level. This approach leads to a new type of organizations, so-called networked and virtual enterprises, where possibly a great number of companies work together without having direct contact but through a broker, as an intermediary, that streamlines the relationships between them. To enable high level efficiency, as well as some other functional requirements, the meta-organizations and brokering services are conceived as environments and services for networked and virtual enterprises operation and dynamic reconfigurations, representing a model of organizations-of-organizations, as an implementation of one of the Industry 4.0 models and ecosystem for networked and virtual enterprises dynamic reconfiguration. In this paper, the meta-organizations with embedded brokering services, modelled as call centres, are analyzed. Various simulations are presented, based on Erlang's formulas for some of design and performance measures parameters evaluation, such as service level, average waiting time, agent occupancy and service traffic intensity.

Asymptotics of insensitive load balancing and blocking phases

We study a single class of traffic acting on a symmetric set of processor-sharing queues with finite buffers, and we consider the case where the load scales with the number of servers. We address the problem of giving robust performance bounds based on the study of the asymptotic behaviour of the insensitive load balancing schemes, which have the desirable property that the stationary distribution of the resulting stochastic network depends on the distribution of job-sizes only through its mean. It was shown for small systems with losses that they give good estimates of performance indicators, generalizing henceforth Erlang formula, whereas optimal policies are already theoretically and computationally out of reach for networks of moderate size. We characterize the response of symmetric systems under those schemes at different scales and show that three amplitudes of deviations can be identified according to whether $$\rho < 1$$ , $$\rho = 1$$ , or $$\rho > 1$$ . A central limit scaling takes place for a sub-critical load; for $$\rho =1$$ , the number of free servers scales like $$n^{ {\theta \over \theta +1}}$$ ( $$\theta $$ being the buffer depth and n being the number of servers) and is of order 1 for super-critical loads. This further implies the existence of different phases for the blocking probability. Before a (refined) critical load $$\rho _c(n)=1-a n^{- {\theta \over \theta +1}}$$ , the blocking is exponentially small and becomes of order $$ n^{- {\theta \over \theta +1}}$$ at $$\rho _c(n)$$ . This generalizes the well-known quality-and-efficiency-driven regime, or Halfin—Whitt regime, for a one-dimensional queue and leads to a generalized staffing rule for a given target blocking probability.

Asymptotics of Insensitive Load Balancing and Blocking Phases

Load balancing with various types of load information has become a key component of modern communication and information systems. In many systems, characterizing precisely the blocking probability allows to establish a performance trade-off between delay and losses. We address here the problem of giving robust performance bounds based on the study of the asymptotic behavior of the insensitive load balancing schemes when the number of servers and the load scales jointly. These schemes have the desirable property that the stationary distribution of the resulting stochastic network depends on the distribution of job sizes only through its mean. It was shown that they give good estimates of performance indicators for systems with finite buffers, generalizing henceforth Erlang's formula whereas optimal policies are already theoretically and computationally out of reach for networks of moderate size. We study a single class of traffic acting on a symmetric set of processor sharing queues with finite buffers and we consider the case where the load scales with the number of servers.We characterize the response of symmetric systems under those schemes at different scales and show that three amplitudes of deviations can be identified according to whether ρ &lt; 1, ρ = 1, and ρ &gt; 1. A central limit scaling takes place for a sub-critical load;for ρ=1, the number of free servers scales like n {θ/θ+1} (θ being the buffer depth and n being the number of servers) and is of order 1 for super-critical loads. This further implies the existence of different phases for the blocking probability. Before a (refined) critical load ρ c (n)=1-a n - {θ/θ+1} , the blocking is exponentially small and becomes of order n - {θ/θ+1} at ρ c (n). This generalizes the well-known Quality and Efficiency Driven (QED) regime or Halfin-Whitt regime for a one-dimensional queue, and leads to a generalized staffing rule for a given target blocking probability.

Traffic-Aware Network Planning and Green Operation with BS Sleeping and Cell Zooming

The traffic load of cellular networks varies in both time and spatial domains, causing many base stations (BS) to be under-utilized. Assisted by cell zooming, dynamic BS sleep control is considered as an effective way to improve energy efficiency during low traffic hours. Therefore, how densely the BSs should be deployed with cell zooming and BS sleeping is an important issue. In this paper, we explore the energy-optimal cellular network planning problem with dynamic BS sleeping and cell zooming for the cases in which traffic is uniformly distributed in space but time-varying. To guarantee the quality of multi-class services, an approximation method based on Erlang formula is proposed. Extensive simulations under our predefined scenarios show that about half of energy consumption can be saved through dynamic BS sleeping and power control. Surprisingly, the energy-optimal BS density we obtained is larger than the one without considering BS sleeping. In other words, deploying more BSs may help to save energy if dynamic BS sleeping is executed.

Application of Erlang Formulae in Next Generation Networks

The paper deals with the possibility of the Erlang B and Erlang C formula utilization in Next Generation Networks (NGN). Based on the common properties of synchronous and asynchronous networks it is possible the utilization of Erlang formulas also for asynchronous networks. It is possible to describe traffic in NGN networks by calculation of following parameters - loss, link utilization and bandwidth. The second part of this paper deals with the possibility of application of Erlang B formula in contact center. Contact center is one of the many examples of the NGN networks. By application of Erlang B formula in contact center environment it is possible to monitor following important parameters - number of agents, probability of call blocking, utilization of agents and mean number of calls in the system.

Internet and the Erlang formula

We demonstrate that the Internet has a formula linking demand, capacity and performance that in many ways is the analogue of the Erlang loss formula of telephony. Surprisingly, this formula is none other than the Erlang delay formula. It provides an upper bound on the probability a flow of given peak rate suffers degradation when bandwidth sharing is max-min fair. Apart from the flow rate, the only relevant parameters are link capacity and overall demand. We explain why this result is valid under a very general and realistic traffic model and discuss its significance for network engineering.

Usage of Erlang Formula in IP Networks

This paper is concerning of Erlang formulas practical approval and their usage with video flow transfer in IP networks. Erlang formulas were simulated in MATLAB program. Practical verification was realized in defined environment by using of designed video source. The result is comparison of loss calculated by MATLAB and loss of real video data transfer.

Towards an Erlang formula for multiclass networks

Consider a multiclass stochastic network with state-dependent service rates and arrival rates describing bandwidth-sharing mechanisms as well as admission control and/or load balancing schemes. Given Poisson arrival and exponential service requirements, the number of customers in the network evolves as a multi-dimensional birth-and-death process on a finite subset of ? k . We assume that the death (i.e., service) rates and the birth rates depending on the whole state of the system satisfy a local balance condition. This makes the resulting network a Whittle network, and the stochastic process describing the state of the network is reversible with an explicit stationary distribution that is in fact insensitive to the service time distribution. Given routing constraints, we are interested in the optimal performance of such networks. We first construct bounds for generic performance criteria that can be evaluated using recursive procedures, these bounds being attained in the case of a unique arrival process. We then study the case of several arrival processes, focusing in particular on the instance with admission control only. Building on convexity properties, we characterize the optimal policy, and give criteria on the service rates for which our bounds are again attained.

Asymptotic Blocking Probabilities in Loss Networks with Subexponential Demands

The analysis of stochastic loss networks has long been of interest in computer and communications networks and is becoming important in the areas of service and information systems. In traditional settings computing the well-known Erlang formula for blocking probabilities in these systems becomes intractable for larger resource capacities. Using compound point processes to capture stochastic variability in the request process, we generalize existing models in this framework and derive simple asymptotic expressions for the blocking probabilities. In addition, we extend our model to incorporate reserving resources in advance. Although asymptotic, our experiments show an excellent match between derived formulae and simulation results even for relatively small resource capacities and relatively large values of the blocking probabilities.

Asymptotic Blocking Probabilities in Loss Networks with Subexponential Demands

The analysis of stochastic loss networks has long been of interest in computer and communications networks and is becoming important in the areas of service and information systems. In traditional settings computing the well-known Erlang formula for blocking probabilities in these systems becomes intractable for larger resource capacities. Using compound point processes to capture stochastic variability in the request process, we generalize existing models in this framework and derive simple asymptotic expressions for the blocking probabilities. In addition, we extend our model to incorporate reserving resources in advance. Although asymptotic, our experiments show an excellent match between derived formulae and simulation results even for relatively small resource capacities and relatively large values of the blocking probabilities.

Insensitive Traffic Models for Communication Networks

We present a survey of traffic models for communication networks whose key performance indicators like blocking probability and mean delay are independent of all traffic characteristics beyond the traffic intensity. This insensitivity property, which follows from that of the underlying queuing networks, is key to the derivation of simple and robust engineering rules like the Erlang formula in telephone networks.

Efficient blocking probability computation of complex traffic flows for network dimensioning

Computing or estimating link blocking probabilities is a fundamental ingredient in network design and engineering. While in traditional telephone networks this was easily done by Erlang's formula, it became much harder in today's complex networks that carry very heterogeneous traffic. We present a simple, efficient method to estimate the blocking probability and link utilization for general multirate, heterogeneous traffic, where the individual bandwidth demands may aggregate in complex, nonlinear ways. The estimation is derived without adopting conventional performance modeling assumptions, such as Poisson arrivals or exponential holding times, thus allowing non-standard behaviour patterns, including special phenomena of IP networks, such as self-similarity. Despite its generality, our estimation maintains an optimally tight exponent and is capable of handling apparently very different cases in a unified way. The approach also makes it possible to make estimations under incomplete information. We show that the results are easily applicable for fast, robust link dimensioning, especially in case of complex traffic patterns, under partial information. Moreover, it is very well fitted for embedding into network level optimization tasks, due in part to simplicity and in part to convexity properties.

The Erlang model with non-poisson call arrivals

The Erlang formula is known to be insensitive to the holding time distribution beyond the mean. While calls are generally assumed to arrive as a Poisson process, we prove that it is in fact sufficient that users generate sessions according to a Poisson process, each session being composed of a random, finite number of calls and idle periods. A key role is played by the retrial behavior in case of call blocking. We illustrate the results by a number of examples.

Dynamic online QoS routing schemes: Performance and bounds

Several dynamic QoS routing techniques have been recently proposed for new IP networks based on label forwarding. However, no extensive performance evaluation and comparison is available in the literature. In this paper, after a short review of the major dynamic QoS routing schemes, we analyze and compare their performance referring to several networks scenarios. In order to set an absolute evaluation of the performance quality we have obtained the ideal performance of any routing scheme using a novel and flexible mathematical programming model that assumes the knowledge of arrival times and duration of the connections offered to the network. This model is based on an extension of the maximum multi-commodity flow problem. Being an integer linear programming model, its complexity is quite high and its evaluation is constrained to networks of limited size. To overcome the computational complexity we have defined an approximate model, based on the multi-class Erlang formula and the minimum multi-commodity cut problem, that provides an upper bound to the routing scheme performance. The performance presented in the paper, evaluated by measuring the connection rejection probability, shows that the schemes considered reach, in several scenarios, the ideal performance, showing that no much gain is left for alternate new schemes.

Managing uncertainty in call centres using Poisson mixtures

AbstractWe model a call centre as a queueing model with Poisson arrivals having an unknown varying arrival rate. We show how to compute prediction intervals for the arrival rate, and use the Erlang formula for the waiting time to compute the consequences for the occupancy level of the call centre. We compare it to the current practice of using a point estimate of the arrival rate (assumed constant) as forecast. Copyright © 2001 John Wiley &amp; Sons, Ltd.

Pure and modified base-stock policies for the lost sales inventory system with negligible set-up costs and constant lead times

The studied inventory system with continuous review has an easily computed optimal ( S−1, S) policy when unsatisfied demands are backlogged. We assume that unsatisfied demands are lost and then it is also easy to compute the best ( S−1, S) policy. But, as demonstrated by Roger Hill at the ISIR Symposium in 1996, this pure base-stock policy can never be optimal if S⩾2. Our focus is on periodic review. We use Erlang's loss formula to derive approximate expressions for the stockout probability and the average cost. These expressions are used to approximate the average cost and to compute a good base-stock. We formulate and implement a Markov decision model to find the optimal replenishment policy. The model is solved by a policy-iteration algorithm. Because the optimal policy is often rather complicated, we introduce modified base-stock policies. They are specified by a pair ( S, t) where S is the base-stock and t is a lower bound for the number of review periods between review epochs in which placing a replenishment order is permitted. A simple one has S equal to the base-stock computed from Erlang's formula and fixes t as the largest integer which is less than or equal to the ratio of the number of review periods per delivery period and S. Our numerical examples show that the simple modified base-stock policy provides most of the cost reduction which can be obtained by replacing the best pure base-stock policy by the optimal policy.

The infinite server queue and heuristic approximations to the multi-server queue with and without retrials

In this paper, properties of the time-dependent state probabilities of theM t /G/∞ queue, when the queue is assumed to start empty are studied. Those results are compared with corresponding time-dependent results for theM/M/1 queue. Approximation to the time-dependent state probabilities of theM/G/m/m queue by means of the corresponding time-dependent state probabilities of theM/G/∞ queue are discussed. Through a decomposition formula it is shown that the main performance characteristics of the ergodicM/M/m/m+d queue are sums of the corresponding random variables for the ergodicM/M/m/m andM/M/1/1+(d−1) queues, respectively, weighted by the 3-rd Erlang formula (stationary probability of waiting or being lost for theM/M/m/m+d queue). Successful exact and approximation extensions of this kind of decomposition formula to theM/M/m/m+d queue with retrials are presented.