Erlang Fixed-point Approximation Research Articles

Revisiting Stochastic Loss Networks: Structures and Approximations

We consider fundamental properties of stochastic loss networks, seeking to improve on the so-called Erlang fixed-point approximation. We propose a family of mathematical approximations for estimating the stationary loss probabilities and show that they always converge exponentially fast, provide asymptotically exact results, and yield greater accuracy than the Erlang fixed-point approximation. We further derive structural properties of the inverse of the classical Erlang loss function that characterize the region of capacities that ensures a workload is served within a set of loss probabilities. We then exploit these results to efficiently solve a general class of stochastic optimization problems involving loss networks. Computational experiments investigate various issues of both theoretical and practical interest, and demonstrate the benefits of our approach.

Performance Modeling of Diversity Coded Path Protection in OBS/OPS Networks

In bufferless optical burst/packet switched (OBS/OPS) networks, data (bursts or packets) may be lost due to contention or equipment failure. Diversity coding-based path protection schemes can be used to protect data from a single-trunk failure with potentially more efficient resource utilization compared to dedicated (e.g., 1 + 1) path protection schemes and help reduce the burst/packet loss ratio. This paper provides a scalable and accurate burst/packet loss ratio approximation based on the Erlang fixed-point approximation for networks that employ protection based on diversity coding and for networks that employ protection based on both diversity coding and 1 + 1 path protection. We use discrete event simulations to assess the accuracy of the approximation based on a wide range of scenarios in a 10-node circular lattice and NSFNet networks. Further, we discuss the effect of parameter settings and the effect of the choice of the wavelength selection method on the accuracy of the approximation. We consider scenarios without wavelength conversion and with full wavelength conversion and two types of users: premium and regular. The premium users, typically of mission critical services, receive protection while the regular users do not. We compare diversity coding and 1 + 1 path protection in a 10-node circular lattice network. The results show that for this example network, for the premium users and under low traffic load, the burst/packet loss ratio in OBS/OPS networks is lower for diversity coding compared to 1 + 1 path protection. However, under heavy traffic load, we observe a lower loss ratio for the case of 1 + 1 path protection.

Performance analysis of green cellular networks with selective base-station sleeping

Base station (BS) sleeping is one of the emerging solutions for energy saving in cellular networks. It saves energy by selectively switching under-utilized BSs to a low power consuming mode (“sleep mode”) during low traffic hours while transferring their associated traffic to active BSs nearby. However, while saving energy, BS sleeping causes a reduction in total available capacity of the network, so Grade of Service (GoS) might be degraded, resulting in a trade-off between energy saving and network performance. This paper proposes a robust and computationally efficient analytical approximation technique, which we call Information Exchange Surrogate Approximation for Cellular Networks (IESA-CN), based on the recently established IESA framework for evaluation of GoS, as measured by call blocking probability, in cellular networks with different BS sleeping patterns. By considering the mutual overflow effect between BSs, the newly proposed method is verified by extensive and statistically reliable simulation experiments to significantly improve the accuracy as compared to traditional Erlang Fixed-Point Approximation in a wide range of scenarios.

Evaluation of Burst/Packet Loss Ratio in a Bufferless OBS/OPS Network With <inline-formula> <tex-math notation="LaTeX">$1+X$ </tex-math> </inline-formula> Path Protection

The importance of avoiding burst/packet loss because of equipment failures, especially for critical-mission services, gives rise to a method known as $1+X$ path protection in optical networks. This method provides additional protection for premium (including emergency) services than the more common $1+1$ path protection. In this letter, we consider a bufferless optical burst switched (OBS)/optical packet switched (OPS) network with two types of users: premium (that receive $1+X$ protection service) and regular (that do not receive such a service). We propose an analytical method based on Erlang fixed-point approximation to evaluate the burst/packet loss of such OBS/OPS network. We demonstrate numerically the accuracy of the approximation, and also the effect and implications of the proportion of the premium traffic as $X$ increases.

Performance Analysis of Circuit Switched Multi-Service Multi-Rate Networks With Alternative Routing

We consider a circuit-switched multiservice network with non-hierarchical alternative routing and trunk reservation. Based on the fundamental concept of overflow priority classification approximation (OPCA), we develop two approximations for the estimation of the blocking probability: OPCA and service-based OPCA. We also apply the classical Erlang fixed-point approximation (EFPA) for the estimation of the blocking probability in the network and propose the more conservative max (EFPA, service-based OPCA) as a reasonably accurate evaluation. We compare the approximations with simulation results and discuss the accuracy of the blocking probabilities for the various traffic classes under different system parameter values such as service rates, bandwidth requirements, number of channels per trunk, maximum allowable number of alternative paths and trunk reservation. We also discuss the robustness of the approximations to the shape of the holding time distribution and their performances under asymmetrical cases. We illustrate that when the approximations are used for network dimensioning, their error is acceptable. We further demonstrate that the approximations can be applied in large networks such as the CORONET.

Blocking Probability Analysis of Circuit-Switched Networks With Long-Lived and Short-Lived Connections

We consider a circuit-switched network with nonhierarchical alternate routing and trunk reservation involving two types of connections that are modeled as long-lived and short-lived calls. The long-lived calls can be reserved well in advance, and the short-lived calls are provided on demand. Therefore, we assume that the long-lived calls have strict priority over the short-lived ones. We develop approximations for the estimation of the blocking probability based on the quasi-stationary approach in two ways. One uses the Erlang fixed-point approximation (EFPA), and the other uses the overflow priority classification approximation (OPCA). We compare the results of the approximations with simulation results and discuss the accuracy of the approximations under different system parameters, such as ratio of offered load, number of links per trunk, maximum allowable number of deflections, and trunk reservation. We also discuss the robustness of the quasi-stationary approximation to the ratio of the mean holding times of the long-lived and short-lived calls as well as that of EFPA and OPCA to the shape of the holding time distribution. Finally, we demonstrate that OPCA can be applied to a large network such as the Coronet.

On the Accuracy of the OPC Approximation for a Symmetric Overflow Loss Model

The overflow priority classification approximation (OPCA) and Erlang's fixed-point approximation (EFPA) are distinct methods for estimating blocking probabilities in overflow loss networks. Mounting numerical evidence has indicated that OPCA provides superior accuracy than EFPA in many circumstances. Furthermore, it has been proven that P EFPA ≤ P OPCA for a symmetric overflow loss network called the distributed server model, where P x is the blocking probability estimate yielded by approximation x ∈ {EFPA, OPCA}. The distributed server model is an ideal “proving ground” because the exact blocking probability, P exact, can be calculated with the Erlang B formula, yet the state dependencies caused by mutual overflow are retained. The present article proves the OPC Conjecture, P OPCA ≤ P exact. This new result establishes that OPCA is always equally or more accurate relative to EFPA for the distributed server model and suggests OPCA has utility in more general overflow loss networks. The proof of P OPCA ≤ P exact turned out to be challenging, since OPCA is remarkably close to the exact solution, requiring delicate inequality arguments on the Maclaurin series of both solutions. Another contribution is the derivation of new tight bounds and limiting regimes for the blocking probabilities yielded by OPCA and EFPA in the case of critical loading. The simple and tight lower bound derived for OPCA can serve as a tight lower bound for the Erlang B blocking probability in the case of critical loading.

Computation of Blocking Probability for Large Circuit Switched Networks

The ever-growing Internet and the mounting evidence to the important role of circuit switching motivate the need for an accurate and scalable means for performance evaluation of large circuit switched networks. Previous work has shown that the Erlang Fixed Point Approximation (EFPA) achieves accurate results for such networks where the number of channels (circuits) per link is large. However, a conventional application of EFPA for large networks is computationally prohibitive. In cases where the EFPA solution is unattainable, we propose, in this paper, to use an asymptotic approximation, which we call A-EFPA, for the link blocking probability and demonstrate savings of many orders of magnitudes in computation time for blocking probability approximation in realistically sized networks with large number of circuits per link. We demonstrate for NSFNet and Internet2 accurate calculations of the blocking probability using simulations, EFPA and A-EFPA, where each of these three methods is used for a different range of parameter values.

Provisioning for large scale loss network systems with applications in cloud computing

Provisioning for large loss networks is a classic problem in performance, due to the fat that loss network is an important mathematical model for many applications, notably those in telephony. Lately, loss network models are utilized and extended to provide performance analysis and control for exciting new applications in statistical physics [3], workforce management [9] and cloud computing [4]. In these new studies, a loss network often serves as a crucial element in characterizing system dynamics and producing calculation for vital performance metrics. This can be seen from the application in resource provision for cloud computing. Cloud computing is rapidly gaining momentum as a new paradigm for offering computing as services via the Internet. Service provider usually offers a menu of service instances, which require the commitment of distinct resources(CPU, Memory,etc) at various amounts. Along with purchase of these instances, service level agreements (SLA) will specify the desired targets on various performance metrics that the service provider should meet. A common performance metric is service availability, defined as the percentage of time at which new service requests can be admitted into the system with their desired amount of resources fulfilled. Violation of the SLAs typically results in significant penalty. The objective of resource provisioning is to seek the balance between the resource costs and SLA penalty so that service availability can be guaranteed efficiently. We develop an integrated optimization framework to search for the optimal resource provision with SLA constraints. First, we develop a Markovian model to capture users’ flexibility on upgrade/downgrade services on demand and characterize the steady-state behavior of the offered load. Then, the multi-class multi-resource provisioning problem can be naturally mapped to a stochastic loss network model, and SLA constraints are mapped to constraints on loss probabilities. Based on Kelly’s approach for capacity planning in a loss network [6, 7, 8], we propose an optimization framework to determine resource levels that minimize the combined costs of resource and violation penalty. Since computing the exact loss probabilities is a ]P complete problem thus prohibitive for large service loads, we consider the Erlang fixed-point approximation for the blocking probabilities that has been proven to be asymptotically exact in the limiting regime as the traffic intensities and the resource capacities grow together in proportion [10, 6, 7]. We further

Admission control and handover management for high‐speed trains in vehicular geostationary satellite networks with terrestrial gap‐filling

AbstractFollowing a recent upgrade, the Digital Video Broadcasting—Return Channel Satellite (DVB‐RCS) standard sets up to support terminal mobility. In this scenario, integration with terrestrial systems becomes a primary concern to ensure network connectivity in urban areas. This article proposes an integrated satellite–terrestrial architecture for the provision of broadband services onboard high‐speed trains, in which terrestrial cellular networks are seen as viable gap‐fillers for discontinuous satellite coverage. We derive an analytical model of the hybrid DVB‐RCS‐cellular system by exploiting analogies between the mobility pattern predictability of LEO constellations and that of high‐speed trains. Terminals whose QoS cannot be guaranteed by the satellite segment are proposed to temporarily divert the connections towards the terrestrial infrastructure, where available. Using an iterative approach based on the Erlang fixed‐point approximation, we show performance improvements with respect to stand‐alone satellite systems in terms of handover failure probability and overall resource utilization. The analytical model is also validated via our ns2‐based DVB‐RCS packet‐level simulator. Detailed modelling of synchronization and signalling mechanisms confirms the accuracy of the analytical results, and shows that topology and mobility information can contribute to refine radio resource utilization optimality when used jointly. Copyright © 2009 John Wiley & Sons, Ltd.

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.

Modeling bufferless packet-switching networks with packet dependencies

In this paper we present a stochastic network model for packet-switching networks with no buffering capabilities at the nodes. This model can be directly used in the study of all-optical packet-switching (OPS) and optical burst switching (OBS) networks without fiber delay lines (FDLs). Our model provides for the first time a complete description of the dependencies arising among packets from different sources in the network. Such dependencies originate when packets from different sources share a finite number of channels for their transmission through a network link. We relate our model to well-known loss network models for circuit-switching networks and derive expressions for the packet loss probability. We briefly show how our work can be extended to model parallel hybrid optical networks, and propose some promising future lines of work. Our numerical results suggest that the well-known Erlang fixed-point approximation (EFPA) overestimates the blocking probability when compared to our model predictions. They also show that our model is scalable up to network scenarios with at least 30 links with 160 wavelength channels per link. This makes the proposed model an interesting tool for studying the dependencies arising among packets in a realistically-sized OPS/OBS network without FDLs.

Analysis of multi-hop traffic grooming in WDM mesh networks

Traffic grooming is an essential functionality of WDM optical networks to provision multi-granularity subwavelength connections. Depending on the number of lightpaths allowed in a connection route, traffic grooming can be classified as single-hop traffic grooming (SH-TG) and multi-hop traffic grooming (MH-TG). MH-TG is more general and resource-efficient than SH-TG, because it allows connections from different source–destination pairs to share the bandwidth of a lightpath. In this paper, we propose a MH-TG algorithm, namely the fixed-order multi-hop (FOMH) grooming algorithm, based on the fixed-alternate routing approach. We introduce the grooming node selection (GNS) problem in MH-TG and propose three grooming policies, namely exhaustive sequential (ES), limited-hop sequential (LHS) and load sharing (LS) policies, to address the GNS problem. These policies represent different trade-offs among blocking probability, computational complexity and transceiver requirements. Given that the analysis of MH-TG is a relatively unexplored area, we propose an analytical model to evaluate the blocking performance of MH-TG using FOMH and the LS grooming policy. To address the multi-layered routing and multi-rate connection characteristics of traffic grooming, we introduce a novel multi-level decomposition approach in our analytical model which decomposes traffic at four different levels, namely alternate path, connection route, lightpath and link levels. The model also addresses various factors that affect connection blocking probability. These factors include wavelength continuity constraint, channel continuity constraint and route dependence. The Erlang fixed-point approximation method is used to solve the analytical model. Numerical results show that analytical results match well with simulation results. We also evaluate the effect of the grooming policies, the number of virtual hops (lightpaths) within a connection route and the number of alternate paths on the performance of the grooming algorithm.

Analytical Performance Computation for all Optical Networks with Wavelength Conversion

In this paper, we have proposed an analytical model that computes the blocking probability and the channel utilisation for a given load on the optical networks with wavelength conversion at the nodes. The computation has been carried out for fixed routing with uniform traffic distribution (UTD). When any link on a route blocks a call, the dropped call reduces the offered load on all other links invovled in the route. The overall blocking probability of the network is estimated using Erlang fixed point approximation with the reduced load on the links. The expression has been validated by comparing the results with that of simulation for a number of standard networks including 14 node NSFNET, 20 node ARPANET and 20 node INDIANET. We show that the analytical method performs well in the desired range of blocking probabilities and is applicable to any network topology.

Enhanced Blocking Probability Evaluation Method for Circuit-Switched Trunk Reservation Networks

We consider the problem of estimating steady-state blocking probability for circuit-switched networks that allow alternate routing with particular emphasis on networks protected by trunk reservation. We use a recently proposed blocking probability estimate, the Overflow Priority Classification Approximation (OPCA) as an alternative to the currently used Erlang's Fixed-Point Approximation (EFPA). We demonstrate empirically that OPCA provides a better blocking estimate than EFPA for circuit-switched networks with alternate routing, that are reasonably protected by trunk reservation.

A new method for approximating blocking probability in overflow loss networks

In this paper, we present a new approximation for estimating blocking probability in overflow loss networks and systems. Given a system for which an estimate of blocking probability is sought, we first construct a second system to act as a surrogate for the original system. Estimating blocking probability in the second system with Erlang's fixed point approximation (EFPA) provides a better estimate for blocking probability in the original system than if we were to use the conventional approach of directly using EFPA in the original system. We present a combination of numerical and theoretical results that indicate our new approximation offers a better estimate than EFPA for a certain pure overflow loss network. Moreover, we demonstrate the accuracy of our new approximation for circuit-switched networks using alternative routing. We argue that the success of our new approximation is due to its ability to utilize congestion information imbedded in overflow traffic, whereas the conventional approach fails to utilize such information.

A Reduced Load Approximation Accounting for Link Interactions in a Loss Network

This paper is concerned with evaluating the performance of loss networks. Accurate determination of loss network performance can assist in the design and dimensioning of telecommunications networks. However, exact determination can be difficult and generally cannot be done in reasonable time. For these reasons there is much interest in developing fast and accurate approximations. We develop a reduced load approximation which improves on the famous Erlang fixed point approximation (EFPA) in a variety of circumstances. We illustrate our results with reference to a range of networks for which the EFPA may be expected to perform badly.

Product form approximations for highly linear loss networks with trunk reservation

Admission controls, such as trunk reservation, are often used in loss networks to optimise their performance. Since the numerical evaluation of performance measures is complex, much attention has been given to finding approximation methods. The Erlang Fixed-Point (EFP) approximation, which is based on an independent blocking assumption, has been used for networks both with and without controls. Several more elaborate approximation methods which account for dependencies in blocking behaviour have been developed for the uncontrolled setting. This paper is an exploratory investigation of extensions and synthesis of these methods to systems with controls, in particular, trunk reservation. In order to isolate the dependency factor, we restrict our attention to a highly linear network. We will compare the performance of the resulting approximations against the benchmark of the EFP approximation extended to the trunk reservation setting. By doing this, we seek to gain insight into the critical factors in constructing an effective approximation.

A reduced load approximation accounting for link interactions in a loss network

This paper is concerned with evaluating the performance of loss networks. Accurate determination of loss network performance can assist in the design and dimen- sioning of telecommunications networks. However, exact determination can be difficult and generally cannot be done in reasonable time. For these reasons there is much interest in developing fast and accurate approximations. We develop a reduced load approximation that improves on the famous Erlang fixed point approximation (EFPA) in a variety of circumstances. We illustrate our results with reference to a range of networks for which the EFPA may be expected to perform badly.

Sequential iteration of the Erlang fixed-point equations

The link or route blocking probabilities of a loss network are typically used to assess its performance. Unfortunately, closed form expressions for these, whilst being easy to write down, are quite intractable to evaluate computationally. Consequently, a number of approximations to the blocking probabilities have been proposed. One of the most intensively studied of these is the Erlang fixed-point approximation. We study the dynamical behaviour of performing sequential iteration on the Erlang fixed-point equations and prove that, for an arbitrary fixed-routing topology with constant bandwidth on all routes, sequential iteration converges to the Erlang fixed point.