Erlang capacity and uniform approximations for shared unbufferedresources

We consider a basic teletraffic model, which has applications to integrated multirate services on ATM and wireless systems. In the “finite-sources” version of the model an unbuffered resource with C channels is shared by heterogeneous sources which alternate between arbitrarily distributed random periods in the on and off states, and in the on state require a fixed number of channels. If a source does not find enough free channels when it turns on, then it is blocked and the burst is lost. In the “infinite-sources” version of the model requests for connections form Poisson streams and connections hold fixed numbers of channels for random periods. The stationary distribution of the system has the product-form and the insensitivity property. Our main results for the finite-sources model are for the asymptotic scaling in which C and the number of sources of each type are large. The central result is a uniform asymptotic approximation (UAA) for the blocking probabilities. It is uniformly effective for the complete range of loadings, simple to calculate and gives accurate results even for relatively small systems. The UAA is also specialized to the overloaded, critical and underloaded regimes. For the admission control of the system we calculate its Erlang capacity, i.e., the set of combinations of sources of various types such that the blocking probabilities for all types do not exceed specified values. For the first two regimes we obtain the boundaries of the admissible sets in the form of hyperplanes, and thus the effective bandwidths of sources of each type. For the underloaded regime the boundary is nonlinear and we obtain a convenient parameterized characterization. Finally, various numerical results are presented     

Performance analysis of a random packet selection policy formulticast switching

This paper studies a random packet selection policy for multicast switching. An input packet generates a fixed number of primary copies plus a random number of secondary copies. Assuming a constant number of contending packets during a slot, the system is modeled as a discrete time birth process. A difference equation describing the dynamics of this process is derived, the solution of which gives a closed form expression for the distribution of the number of packets chosen. Then this result is extended to the steady state distribution through a Markov chain analysis. It is shown that the old packets have larger fanout than the fresh packets and the copy distribution of the mixed packets is determined. The packet and copy throughput taking into account the old packets have been obtained. We determined the mean packet delay as well as an upperbound for packet loss probabilities for finite buffer sizes. The asymptotic distribution of the number of packets is also given for large switch sizes under saturation by applying results from the renewal theory. Finally, simulations are done to determine the performance of the switch under mixed (unicast plus multicast) traffic     

New call blocking versus handoff blocking in cellular networks

In cellular networks, blocking occurs when a base station has no free channel to allocate to a mobile user. One distinguishes between two kinds of blocking, the first is called new call blocking and refers to blocking of new calls, the second is called handoff blocking and refers to blocking of ongoing calls due to the mobility of the users. In this paper, we first provide explicit analytic expressions for the two kinds of blocking probabilities in two asymptotic regimes, i.e., for very slow mobile users and for very fast mobile users, and show the fundamental differences between these blocking probabilities. Next, an approximation is introduced in order to capture the system behavior for moderate mobility. The approximation is based on the idea of isolating a set of cells and having a simplifying assumption regarding the handoff traffic into this set of cells, while keeping the exact behavior of the traffic between cells in the set. It is shown that a group of 3 cells is enough to capture the difference between the blocking probabilities of handoff call attempts and new call attempts. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analyzing Product-Form Stochastic Networks Via Factor Graphs and the Sum-Product Algorithm

A large number of stochastic networks including loss networks and certain queueing networks have product-form steady-state probabilities. However, for most practical networks, evaluating the system performance is a difficult task due to the presence of a normalization constant. We propose a new framework based on probabilistic graphical models to tackle this task. Specifically, we use factor graphs to model the stationary distribution of a network. For networks with arbitrary topology, we can apply efficient message-passing algorithms like the sum-product algorithm to compute the exact or approximate marginal distributions of all state variables and related performance measures such as blocking probabilities. Through extensive numerical experiments, we show that the sum-product algorithm returns very accurate blocking probabilities and greatly outperforms the reduced load approximation for loss networks with a variety of topologies. The factor graph model also provides a promising approach for analyzing product-form queueing networks.     

Performance analysis of resource sharing policies in CDMA networks

We consider an interference‐limited code division multiple access (CDMA) network, in which some of the service classes are explicit rate controlled. Such rate‐controlled service classes include adaptive ones whose holding time does not depend on the sending rate, while the holding time of elastic services becomes longer when their sending rate is slowed down. We develop a Markov model that allows us to study the impact of employing service differentiation (prioritization) during admission control and rate control on the classwise blocking probabilities and throughputs. The admission control takes into account the interference from adjacent cells as a log‐normally distributed random variable and employs service class and state‐dependent (soft) blocking in addition to hard blocking. The state‐dependent rate control algorithm takes into account the minimum and maximum requested rates of admitted sessions in a fairness policy‐dependent manner. Our conclusion is that work conserving bandwidth sharing policies may have a major impact on the moments of the time spent in the system by elastic services, but only a minor one on the classwise blocking probabilities. When the admission control algorithm takes into account the neighbour cell caused interference, the blocking probability of all classes increases somewhat, but the throughput of the accepted sessions increases. Blocking prioritization in the soft blocking model affects the classwise total blocking probabilities but has little impact on the throughput of adaptive and elastic traffic. Copyright © 2006 John Wiley & Sons, Ltd.     

A Model for Telephone Networks and Its Use for Routing Optimization Purposes

The problem studied here concerns the modeling of call blocking in telephone networks. From the usual assumptions such as exponential arrivals and holding time, lost call cleared, the state of the network is described by a finite Markov chain. From the transition probabilities of this process are derived the differential equations associated with the average occupancy of all trunk groups. These traffic equations are simplified by considering independence of blocking for trunk groups in series. The blocking probabilities are estimated using fictitious offered traffic and the Erlang B formula. Such representation takes into account peaky or smooth traffic characteristics. We develop this one-moment model for routing policies such as load sharing and overflow routing. Performances of the model are given in comparison to the solution of the exact Markov chain model or the results of Monte Carlo simulation. Finally, an application to routing optimization and network dimensioning is treated.     

A performance model for ATM switches with general packet lengthdistributions

The maximum throughput of an ATM switch is investigated in the presence of an offered load of multicell packets. For the case of input queueing coupled with a round-robin policy for transferring cells from inputs to outputs, the system is approximated by a product form queueing network. Under the assumption that packet lengths are described by random variables with discrete Coxian distributions, it is shown that the balance equations describing the behavior of the ATM switch approach those for a product form queueing network and that the steady-state probabilities of such an ATM switch approach the product-form solution as the cell length tends to zero. Last, a numerical investigation shows that the approximation yields good results, even when the packet lengths are not well described by Coxian distributions     

The maximum throughput of a nonblocking space-division packetswitch with correlated destinations

We propose a discrete-time queueing network model for space-division packet switches, in which the destinations of consecutive packets may be correlated, and different input traffic may have different destination distributions. We show that for switches under saturation, the steady-state probabilities have an approximate product-form solution. The solution is very accurate for real applications where the average packet sizes are often more than 10 times the number of bytes that can be transmitted in one time slot     

Admission-control policies for multihop wireless networks

In this paper, we investigate the admission-control problem for voice traffic in fixed-route circuit-switched wireless networks. We consider coordinate-convex admission-control policies and a blocked-calls-cleared mode of operation, in conjunction with the usual assumptions on the voice process statistics. These conditions result in a product-form stationary distribution for the voice state of the system, which facilitates the evaluation of network performance. However, to determine the optimal policy a large state space must be searched. We develop a recursive procedure to accelerate the evaluation of a large number of different admission-control policies, and a descent-search method to reduce significantly the number of policies that must be evaluated in searching for the optimal one. The numerical examples we present indicate that reduced blocking probability (or increased throughput) can be obtained by administering active admission control. The degree of improvement is highest in moderately overloaded traffic conditions, but it is typically small in low-capacity networks (at all loads). However, in applications where the performance measure associates different revenues or costs with the various call types, considerable improvement can be obtained when admission control is used.     

Blocking in tandem ISDN links

Broadband optical networks which carry a wide range of traffic classes in an integrated fashion are considered. The focus is on so-called stream traffic in which each class if allocated different quantities of broadband for the duration of a call on an equal sharing policy. An exact analysis of the blocking conditions experienced by different classes of traffic demanding transmission through a network with the star or the ring topology is presented. Thus, blocking probabilities on tandem links are calculated by means of a recursive relation that considerably simplifies the complexity of the problem without ignoring the interdependencies among links. Being a solution to a multiple-classes-multiple-links blocking problem, the present work is a multidimensional generalization of a well-known analysis for a single trunk. The efficiency of the solution results from the linearity of the necessary computations     

Blocking Probabilities for Trunk Reservation Policy

This paper presents an analysis of the blocking phenomenon in circuit-switched routing policies with alternate routing and trunk reservation. From the usual assumptions on the offered traffic (Poisson) and the service time (exponential distribution), a classical Markov chain model is derived. Some approximate formulas are given to compute the blocking probabilities for first- and second-choice traffic. The accuracy of the proposed estimates is evaluated by direct comparison with already known results in simple cases and with the exact solution of the state equations.     

Proportional differentiated admission control

This letter presents a new admission control policy inspired in the framework of proportional differentiated services (PDS). While most of previous PDS has focused on average queueing delays and packet drops to differentiate the performance of adaptive applications, the proportional differentiation admission control (PDAC) differentiates inelastic traffic in terms of blocking probabilities. The PDAC is built up using asymptotic approximation theory, employs a class based approach, and conforms with the PDS requirements of predictability and controllability . Numerical experiments confirm a good performance of the approach.     

Fairly adjusted multimode dynamic guard bandwidth admission control over CDMA systems

Guard-based call admission control schemes support admission priorities based on resources sharing with differentiated resource capacity limits. To minimize deviation from call blocking/dropping targets due to nonstationary call arrival condition, dynamic guard-based schemes with predictive adaptation control adjust differentiated capacity limits according to predicted future arrival rates based on specified estimation algorithms. Existing dynamic guard admission schemes are developed under the assumption of perfect estimation, which may not be possible in a highly nonstationary environment and, thus resulting in failures to maintain targeted blocking/dropping probabilities. This paper presents the fairly adjusted multimode-dynamic guard bandwidth scheme, which is a dynamic-guard-based scheme over code-division multiple-access systems with predictive adaptation control to adapt interference-based guard loading-limits under nonstationary call arrival condition; and reactive adaptation control to counteract arrival rate estimation errors. When the predictive adaptation control policy mode is not able to maintain long-term call blocking or dropping targets due to estimation error, this will trigger reactive adaptation control policy modes that include temporary blocking (preemption) of one or more lower priority classes subject to fairness constraints to ensure lower priority classes are not preempted at all costs during estimation error recovery. Analytical and simulation results show that proposed scheme is able to provide performance guarantees in terms of dropping probabilities under nonstationary traffic arrival and imperfect arrival rate estimation.     

Erlang capacity of a TDD-TD/CDMA architecture supportingheterogeneous traffic

This letter determines the capacity of a TDD-TD/CDMA architecture which supports different classes of subscribers and adopts an interference-driven admission control policy. The blocking probabilities of the system users are evaluated under various traffic conditions for several uplink/downlink configurations of the time slots, demonstrating that the time division full duplex approach needs careful tuning in order to maximize system capacity     

Performance of a finite capacity asynchronous multiplexer with modulated input

In this paper, a finite capacity multiplexer in ATM environments with modulated deterministic input sources is investigated. We consider three cases of input traffic: (1) a fixed number of deterministic cell-generating sources, (2) a fixed number of sources where each source is modulated by an ON-OFF process, and (3) a variable number of ON-OFF sources, modulated by a call arrival process in conjunction with a connection admission control scheme. The main performance measures in this study are the cell and call blocking probabilities. For case (1), we give a tight upper bound for the transient phase of the state process to reach stationary conditions. The state probabilities and the corresponding cell blocking probability are determined by means of an exact analysis. For cases (2) and (3), where we deal with more complex input traffic, the compound state process and the overall cell blocking probability are analyzed using a quasi-stationary approximation technique. This approximation is validated by means of simulations. The analysis presented is shown to be able to give reasonably accurate approximate results for ATM-multiplexer performance for cases in which conventional simulation and analytic methods already approach computing time and complexity boundaries.     

A buffered two-node packet radio network with product form solution

A packet radio network with two nodes, both of which access a single radio channel using the slotted ALOHA protocol, is discussed. This model results in a network of two interfering queues. Networks of interfering nodes normally do not have partial balance and thus seldom have product-form solution (PFSs). However, when certain assumptions are made on the network parameters, PFS-controls can be found that yield a PFS for the interfering two-node network. Three such cases are described. In the first two cases, PFS controls are found that control the new packet arrival rate. When operating under these PFS controls, the two-mode network has a simple, closed-form solution of the equilibrium probabilities, the expected throughput, and the expected packet delay. The throughput-delay performance closely approximates that of an exact two-node packet radio model. In the third case, a PFS control is found on one node's transmission probabilities. When operating under this PFS control, the transmission probabilities approach asymptotic values as the packet backlog increases. The network tends to adjust its transmission probabilities to reduce large unequal packet backlogs     

Dimensionnement de bout en bout des réseaux téléphoniques hiérarchisés et combinaison des trafics

The author considers the problem of determining the individual loss probabilities of each traffic stream in a hierarchical network operated as a loss system in the case of arbitrary call holding time distribution. He shows that, in order to separate out the distribution function of each traffic stream, it is necessary to modify not only the arrival processes but also the call holding time distribution functions. The usual concept of «traffic source», which is convenient for isolating a part of the network, no longer applies. Therefore care must be exercised when applying approximate occupancy distribution moment matching methods. «Pseudo — Poisson» traffics are shown to combine in «weighted» sums of pseudo-Poisson traffics. The weighting coefficients are boundary values having different properties and are directly related to end to end blocking probabilities.     

Normal-type approximation for multi-service systems with trunk reservation

In this paper we consider a transmission link of a broadband system. A number of heterogeneous streams with different traffic intensities, different call holding times and different bandwidth requirements use the link in a complete sharing mode with trunk reservation. A normal-type approximation formula for individual blocking probabilities is described. Some numerical examples are presented to show the accuracy of the approximation.     

Algorithms to determine exact blocking probabilities for multiratetree networks

A circuit-switched network consisting of multiple-access links connected to a common link is considered. Each call requires circuits on one access link and on the common link. The network supports multiple classes of calls where each class specifies a bandwidth requirement, an arrival rate, and a holding-time distribution. Based on a product-form solution for these networks, four algorithms are developed to determine the exact blocking probability for each of the classes. The first two algorithms are based on convolution, the third on the fast Fourier transform, and the fourth on a recursion due to J.S. Kaufman (1981) and to J.W. Roberts (1981). Complexity bounds and numerical results demonstrate that these algorithms can determine blocking probabilities in reasonable CPU time for networks with thousands of circuits     

Blocking Bounds for Random Channel Selection on Tree Topologies

Nonloopy network topologies are considered under the limitation that a channel can be utilized only if it is idle at all neighboring sites. Random channel selection is studied under a standard circuit-switched traffic model. Upper and lower bounds for blocking probabilities are determined via an auxiliary network process whose equilibrium distribution admits a computationally convenient form. By way of another approximate characterization, it is argued that random channel selection incurs vanishing loss of optimality as the number of channels and the traffic load increase in proportion.