Search

We consider a multi-class multi-server retrial queueing system. Customers of each class arrive from outside the system according to a Poisson process. The service times for each class of customers are assumed to be exponentially distributed with service rates depending on both the customers’ class and the servers. We define the offered load and provide stability and instability conditions for this retrial queueing system. The stability result can be obtained by introducing artificial primitive processes and using the fluid limit approach.

We consider a queueing system with heterogeneous customers. One class of customers is eager; these customers are impatient and leave the system if service does not commence immediately upon arrival. Customers of the second class are tolerant; these customers have larger service requirements and can wait for service. In this paper, we establish pseudo-conservation laws relating the performance of the eager class (measured in terms of the long run fraction of customers blocked) and the tolerant class (measured in terms of the steady state performance, e.g., sojourn time, number in the system, workload) under a certain partial fluid limit. This fluid limit involves scaling the arrival rate as well as the service rate of the eager class proportionately to infinity, such that the offered load corresponding to the eager class remains constant. The workload of the tolerant class remains unscaled. Interestingly, our pseudo-conservation laws hold for a broad class of admission control and scheduling policies. This means that under the aforementioned fluid limit, the performance of the tolerant class depends only on the blocking probability of the eager class, and not on the specific admission control policy that produced that blocking probability. Our psuedo-conservation laws also characterize the achievable region for our system, which captures the space of feasible tradeoffs between the performance experienced by the two classes. We also provide two families of complete scheduling policies, which span the achievable region over their parameter space. Finally, we show that our pseudo-conservation laws also apply in several scenarios where eager customers have a limited waiting area and exhibit balking and/or reneging behaviour.

Voice over Internet Protocol (VoIP) uses packet switching to transmit voice, video, data and chat services. VoIP applications depend upon various speech codecs which vary in speech quality, bandwidth and computational requirements. The key challenges for improving the VoIP Quality of Service (QoS) and Quality of Experience (QoE) is to use best codec by considering network traffic and selecting the optimal hardware. Hardware Calibration is a mechanism to aid service providers by selecting a suitable hardware for Call Manager to transmit different real-time based applications. The paper focuses on proposing a computational model which suggests the most suitable hardware for a specific load handling requirement. The selection of processor is done by taking into consideration the effect of different codecs when it takes calls per second as input and gives output as whichever hardware is recommended for Call Manager to handle different types of applications like Peer-To-Peer (P2P) application, Back-To-Back (BTB) application, enabling voice logging, etc. The emulations demonstrate that the proposed model is suitable for selection of hardware which provides better QoS and QoE for the transmission VoIP based real-time applications. Further, a multi-criteria decision-making method based on Analytic Hierarchy Process (AHP) is also proposed to decide which hardware can be chosen as Call Manager. The results obtained from AHP are used to validate the outcome of the proposed computational model. The computational model is applicable only when the tests are run on particular systems and it suggests the best hardware among them for the particular offered load. However, to suggest the best configuration corresponding to the input load from the possible configurations available in the market, benchmarking is performed. A common scale model for hardware benchmark estimation particularly has been proposed for different scenarios.

The FDL buffers can have only discrete delay values. Because of this discontinuity, in order to construct the FDL buffers, some parameters such as the offered load, the average data burst length, and the basic delay unit, of which the length of each FDL is consecutive multiples, should be considered. This means that if one or more parameters change, new FDL buffers are required. So, even when one or more parameters change, in order to minimize the effect of the change, a new service differentiation algorithm dynamically controlling data burst length based on a shared-type feed-forward FDL architecture is proposed in this paper. Various results show that the algorithm improves fairness between classes and significantly reduces the fluctuation of the number of delay lines for each class.

When the boundary--the total number of servers-- in an Erlang loss system is a function of time, customers may also be lost due to boundary variations. On condition that these customers are selected independently of their history, we solve for the hitting-time distribution and transient distribution of busy servers. We derive concise asymptotic expressions in the time domain for normal loads in the heavy-traffic limit, i.e., when the offered load $$\rho $$ ? is high, and the number of servers scales as $$\rho +O\left( \sqrt{\rho }\right) $$ ? + O ? . The solutions are computationally efficient, and simulations confirm the theoretical results.

Resource scheduling in large-scale distributed systems, such as grids and clouds, is difficult due to the size, dynamism, and volatility of resources. These resources are eclectic and autonomous, and may exhibit different usage policies, levels of participation, capabilities, local load, and reliability. Moreover, applications are likely to exhibit various patterns and levels, and distributed resources may organize into various different overlay topologies for information and query dissemination. Researchers have proposed a wide variety of approaches and policies for mapping offered load onto resources and for solving the various component parts of the scheduling problem. However, production clouds and grids may be underutilized, and may not exhibit the load to effectively characterize all of the scheduling system inputs. The composition of large-scale systems is also changing, potentially to include more individual and peer-to-peer resources. These factors will influence the effectiveness of proposed scheduling solutions. Therefore, a simulation environment is necessary to study different approaches under different scenarios, especially those that are expected, but that are not currently characteristic of existing systems. This article describes a general-purpose peer-to-peer simulation environment that allows a wide variety of parameters, protocols, strategies and policies to be varied and studied. To provide a proof of concept, utilization of the simulation environment is presented in a large-scale distributed system problem that includes a core model and related mechanisms. In particular, this article presents a definition and possible peer-to-peer solutions for the large-scale scheduling problem. Moreover, this article describes a general simulation model, some policies that can be varied, an implementation, and some sample results.

Scheduling in large scale dynamic grids comprising eclectic collections of resources is increasingly difficult. Autonomous resource neighborhoods may wish to determine the level of grid offered load that they can or will accept; different sites may wish to attract different amounts of load, to satisfy some desired property within a grid economy. This changes the traditional notion of load sharing, which generally assumes that the desired equilibrium should be an equal distribution of load across all participating machines, because they are under the jurisdiction of a single site, and therefore more likely to implement one common policy. In large-scale grids, nodes and neighborhoods should instead get a portion of the load that best matches their local policies for supporting and admitting grid jobs. This article describes information dissemination protocols that can distribute load in this way, without using load rebalancing through job migration, which is more difficult and costly in large-scale heterogeneous grids. Essentially, nodes adjust their advertising rates and aggressiveness to influence where jobs get scheduled. We report experimental results with example resource configurations in which each resource neighborhood determines its ideal grid load and disseminates accordingly. In turn, each neighborhood attracts the requisite amount of resource requests from the grid. Moreover, performance does not degrade: overall query satisfaction rates are within 9% of both adaptive dissemination protocols that use static adaptation policies, and static dissemination protocols that may be custom-tailored to specific resource and load distributions.

In this article we propose the Minimization of Probability and Size of scheduling Latency (MinPSL) protocol for remarkably efficient load balancing and scheduling of variable length messages in WDM star networks. Our EMPS protocol has been shown to noticeably increase the performance of practical networks with non-uniform destinations and non-negligible transceiver tuning latencies, while other well-known schemes like EATS and MSL seem to suffer under these realistic conditions. EMPS tries to balance the offered load by only considering the heads of the message queues maintained by the source nodes. As a result, its performance is degraded, especially for medium to high offered loads and for rather slowly tunable transceivers. The proposed MinPSL scheme, on the other hand, examines each source node’s message queue up to a certain depth and assigns higher priorities to messages intended for the least used destinations. In this way, MinPSL perfectly minimizes the probability of having to schedule two or more messages intended for the same destination simultaneously or at close times. Furthermore, it also minimizes the actual performance penalty incurred, when scheduling of consecutive messages with the same destination cannot be avoided. Extensive simulations are carried out in order to study the performance of MinPSL and compare it to the EMPS scheme under various conditions. The simulation results ascertain the fact that the proposed MinPSL protocol always brings about a considerable performance improvement.

We consider a network control problem for wireless networks with flow level dynamics under the general k-hop interference model. In particular, we investigate the control problem in low load and high load regimes. In the low load regime, we show that the network can be stabilized by a regulated maximal scheduling policy considering flow level dynamics if the offered load satisfies a constraining bound condition. Because maximal scheduling is a general scheduling rule whose implementation is not specified, we propose a constant-time and distributed scheduling algorithm for a general k-hop interference model which can approximate the maximal scheduling policy within an arbitrarily small error. Under the stability condition, we show how to calculate transmission rates for different user classes such that the long-term (time average) network utility is maximized. This long-term network utility captures the real network performance due to the fact that under flow level dynamics, the number of users randomly change so instantaneous network utility maximization does not result in useful network performance. Our results imply that congestion control is unnecessary when the offered load is low and optimal user rates can be determined to maximize users' long-term satisfaction. In the high load regime where the network can be unstable under the regulated maximal scheduling policy, we propose a cross-layer congestion control and scheduling algorithm which can stabilize the network under arbitrary network load. Through extensive numerical analysis for some typical networks, we show that the proposed scheduling algorithm has much lower overhead than other existing queue-length-based constant-time scheduling schemes in the literature, and it achieves performance much better than the guaranteed bound.

We propose a new and exact method to analyze the throughput of the DS CDMA unslotted ALOHA system with variable number of fixed-length packets. The proposed scheme requires two-dimensional summation while the previous requires four. Thus the proposed analysis scheme is of less computational complexity. The proposed analysis result is also slightly closer to the simulation result than the previous analysis result is, especially in heavier offered load.

A high-performance Efficient Message Prioritization and Scheduling (EMPS) protocol, for intelligent message scheduling in Wavelength-Division Multiplexing (WDM) star networks is introduced. The performance of the well-known EATS and MSL schemes is noticeably degraded in practical networks with non-uniform destinations and non-negligible transceiver tuning latencies. Under these realistic conditions, it is common that two or more messages with the same destination have to be scheduled consecutively or at close times. In most cases, this brings about some performance penalty, owing to the delayed availability of the destination’s receiver for the second (and beyond) of the consecutive messages. As the frequency of such occurrences increases, the performance degradation of the existing schemes becomes more prominent. EMPS is proposed to deal with this problem. It simultaneously considers multiple messages from different transmitting nodes and gives priority to messages intended for the least used destinations each time. By balancing the offered load in this way, EMPS minimizes the probability of having to schedule two or more messages with the same destination consecutively or at close times. Additionally, by incorporating the Minimum Scheduling Latency algorithm for channel selection, the protocol also minimizes the actual performance penalty incurred, when scheduling of consecutive messages with the same destination cannot be avoided. Extensive simulations are carried out in order to study the performance of EMPS and compare it to other efficient schemes under various conditions. The simulation results show that the proposed protocol always brings about a significant performance improvement.

The percentages of blocking and forced termination rates as parameters representing quality of services (QoS) requirements are presented. The relation between the connection statistics of mobile users in a cell and the handoff number and new call number in next duration in each cell is explored. Based on the relation, statistic reservation tactics are raised. The amount of bandwidth for new calls and handoffs of each cell in next period is determined by using the strategy. Using this method can guarantee the communication system suits mobile connection request dynamic. The QoS parameters: forced termination rate and blocking rate can be maintained steadily though they may change with the offered load. Some numerical experiments demonstrate this is a practical method with affordable overhead.

A novel scheme, namely united stabilizing scheme for edge delay, is introduced in optical burst switched networks. In the scheme, the limits of burst length and assembly time are both set according to certain qualifications. For executing the scheme, the conception for unit input bit rate is introduced to improve universality, and the assembly algorithm with a buffer safety space under the self-similar traffic model at each ingress edge router is proposed. Then, the components of burst and packet delay are concluded, and the equations that limits of burst length and assembly time should satisfy to stabilize the burst edge delay under different buffer offered loads are educed. The simulation results show that united stabilizing scheme stabilizes both burst and packet edge delay to a great extent when buffer offered load changes from 0.1 to 1, and the edge delay of burst and packet are near the limit values under larger offered load, respectively.

We present a cost model for splitting Internet dial-up traffic (which varies by time-of-day) between two large modem banks. One of the modem banks charges by the hour, the other charges for the peak number-in-system during the day. To study if the possible savings are enough to make the effort worthwhile, we formulate a clairvoyant ("perfect information") Integer Program that is equivalent to a network flow problem. This leads us to use a ceiling policy. In the stochastic control case, we use a Modified Offered Load (MOL) approximation to explore the properties of the system, and develop a square-root-type rule to set the ceiling in the homogeneous case. We also use simulation to determine an optimal ceiling when we cannot route individual calls precisely. We propose approximations that may be computed for any call duration distribution, and compare their answers to exact differential-equation procedures for Exponential call durations.

We analyze spectral efficiency of dynamic time division duplex in a fixed wireless cellular network. Conventionally, spectral efficiency has been analyzed for static and fully loaded systems. In this paper, we investigate how asymmetric and dynamic traffic affect the spectral efficiency of Time-Division-Duplex (TDD) systems. Recently, dynamic TDD (D-TDD) has gained much attention as an efficient duplex scheme for high-speed data communications, because adaptive switching ability enables the system to obtain statistical multiplexing gain by exploiting dynamic and asymmetric data traffic. However, it has been noted that a rather strong co-channel interference can be present due to adaptive switching in a cellular network that uses frequency reuse. Thus, benefits of dynamic TDD may not be justifiable unless a proper countermeasure is employed. To suppress the effect of strong co-channel interference, we employ time slot allocation (TSA) strategy along with sector antenna layouts, as proposed in our previous work Jeong and Kavehrad (IEEE and Transactions on communication, Vol. 50, no.10 pp. 1627–1636, 2002). We note that higher spectral efficiency is obtained in D-TDD systems by employing TSA strategy. We also evaluated spectral efficiency of D-TDD system employing adaptive modulation, assuming that traffic is delay tolerant. It is observed that five times higher spectral efficiency can be obtained by employing adaptive modulation. The effect of variance of ratio of offered load on uplink and downlink is also evaluated. Our computer simulation results show that spectral efficiency of D-TDD system with time slot allocation algorithm is more than that of static TDD (S-TDD) over a large range of traffic variation. In conclusion, D-TDD system can take advantage of statistical multiplexing gain of dynamic traffic by adaptively positioning the boundary to the varying traffic bandwidth in its two-way transmissions when TSA strategy is employed for suppression of strong co-channel interference.

This paper presents a study on dynamic wavelength routed all-optical networks by simulating traffic on all-optical networks. A performance study is carried out on dynamic all-optical networks for fixed and free routing. It is explained how multiple fibers correspond to limited wavelength conversion, and it is explained why the presence of wavelength converters increase the complexity of optical cross connects. We find that both free routing and wavelength conversion lowers the blocking probability significantly. The new contribution is that we determine the gain in blocking probability as function of the number of fibers per link and the offered load. We find that multiple fibers reduce the effect of wavelength converters significantly.

Time dependent behavior has an impact on the performance of telecommunication models. Examples include: staffing a call center, pricing the inventory of private line services for profit maximization, and measuring the time lag between the peak arrivals and peak load for a system. These problems and more motivate the development of a queueing theory with time varying rates. Queueing theory as discussed in this paper is organized and presented from a communications perspective. Canonical queueing models with time-varying rates are given and the necessary mathematical tools are developed to analyze them. Finally, we illustrate the use of these models through various communication applications.

We study the delay in asymmetric cyclic polling models with general mixtures of gated and exhaustive service, with generally distributed service times and switch-over times, and in which batches of customers may arrive simultaneously at the different queues. We show that (1−ρ)X i converges to a gamma distribution with known parameters as the offered load ρ tends to unity, where X i is the steady-state length of queue i at an arbitrary polling instant at that queue. The result is shown to lead to closed-form expressions for the Laplace–Stieltjes transform (LST) of the waiting-time distributions at each of the queues (under proper scalings), in a general parameter setting. The results show explicitly how the distribution of the delay depends on the system parameters, and in particular, on the simultaneity of the arrivals. The results also suggest simple and fast approximations for the tail probabilities and the moments of the delay in stable polling systems, explicitly capturing the impact of the correlation structure in the arrival processes. Numerical experiments indicate that the approximations are accurate for medium and heavily loaded systems.

Call-blocking probabilities are among the key performance measures in mobile communications networks. For their analysis, mobile networks can be modelled as networks of Erlang loss queues with common capacity restrictions dictated by the allocation of frequencies to the cells of the network. However, due to the time-varying load offered to the cells of such networks, blocking probabilities usually cannot be obtained in closed form. The relation between networks of Erlang loss queues and networks of infinite server queues, for which the time-dependent occupancy distribution is multidimensional Poisson, suggests to use that distribution as approximate distribution for the network of Erlang loss queues. This paper extends this so-called Modified Offered Load (MOL) approximation to networks of Erlang loss queues, and also allows subscribers that find their call blocked to redial to continue their call. For GSM networks operating under Fixed Channel Allocation, it is shown that blocking probabilities are increasing in the redial rates so that the MOL approximation that is most accurate for maximal redial rates turns out to be fairly accurate for the resulting upper bound for blocking probabilities. The accuracy is explicitly evaluated in an application of the results towards blocking probabilities in a hot spot travelling along a road through a GSM network.

Motivated by the desire to appropriately account for complex features of network traffic revealed in traffic measurements, such as heavy-tail probability distributions, long-range dependence, self similarity and nonstationarity, we propose a nonstationary offered-load model. Connections of multiple types arrive according to independent nonhomogeneous Poisson processes, and general bandwidth stochastic processes (not necessarily Markovian) describe the individual user bandwidth requirements at multiple links of a communication network during their connections. We obtain expressions for the moment generating function, mean and variance of the total required bandwidth of all customers on each link at any designated time. We justify Gaussian approximations by establishing a central limit theorem for the offered-load process. We also obtain a Gaussian approximation for the time-dependent buffer-content distribution in an infinite-capacity buffer with constant processing rate. The offered-load model can be used for predicting future bandwidth requirements; we then advocate exploiting information about the history of connections in progress.

A reservation-based protocol based on a Pipelining Cyclic Scheduling Algorithm (PCSA) is proposed for packet-switched single-hop photonic networks. This protocol contains a mechanism to avoid contention at the receivers. Packets arrive in order and the transmission delay and its variations are optimized. The effects of propagation delay and processing time are almost compensated for by a pipelining technique. Analytical models and analysis are developed. The transmission delay is calculated as a function of the offered load, the number of nodes, propagation delay and processing time. Finally, the results of the analysis and simulations are compared.

In this paper, we present the design of a large self-routing multicast ATM switch. The switch consists of a sorting network followed by a 3-stage routing network. We first present a simple design of a large sorting network built using small sized shared memory that can be used as a building block for a large sorting network. Small sized shared memory is also used in the 3-stage routing network making the switch modular and easy to implement using current VLSI technology. As the network uses shared memory modules, multicasting functionality is easily built into the network. The performance of the proposed network is compared with an equivalent completely shared memory switch using computer simulations under bursty traffic model. The results show that the proposed network has better performance in terms of cell loss ratio than the completely shared memory switch under moderate to heavy traffic load (0.6 ≤ effective offered load ≤ 1.2). Furthermore, multicast cell delays are drastically improved.

Priority based link-bandwidth partitioning is required to support wireless multimedia services, having diverse QoS (delay, throughput) requirements, in mobile ad hoc networks with multimedia nodes. A new class of service disciplines, termed “batch and prioritize” or BP admission control (AC), is proposed. The BP algorithms use the delay tolerance of applications to batch requests in time slots. Bandwidth assignment is made either at the end of the slot, or during the slot, on a priority basis. Analytical and simulation models are developed to quantify the performance of the BP schemes. The results are compared with those obtained for a First-Come-First-Served (FCFS) service discipline. The class of BP schemes trade off the delay and loss tolerance of applications to improve the net carried traffic on the link. Further, such schemes enable an easy implementation for adaptive prioritization, where the degree of precedence given to an application varies with offered load and the link capacity.

This paper deals with planning system activities to support applications that have different contrasting requirements including timing constraints on tasks execution and correctness requirements. Our approach is based on a simple yet effective formulation of a value structure associated to the application tasks. Values are associated to each relevant outcome thus accounting for successful executions as well as for those which violate the application requirements. Moreover we assume degradable real time systems equipped with several execution techniques characterised by different execution costs and different levels of fulfilment of requirements (and associated reward). We propose an admission algorithm to select the part of the offered load to be executed, should overload occur. For all the admitted tasks the algorithm selects also the most suitable execution technique (among those available) to optimise the expected cumulated reward. We show that the algorithm provides the best solution to the optimisation problem resorting to the linear programming theory. Then we discuss the applicability of this result to systems operating in dynamic environments. A planner component is defined, responsible to collect information on the current status of the system and of its environment. The planner decides when a new ’plan‘ is required, and dynamically executes the admission algorithm to properly tune the usage of system resources.

In optical WDM networks, an assignment of transceivers to channels implies an allocation of the bandwidth to the various network nodes. Intuition suggests, and our recent study has confirmed, that if the traffic load is not well balanced across the available channels, the result is poor network performance. Hence, the time-varying conditions expected in this type of environment call for mechanisms that periodically adjust the bandwidth allocation to ensure that each channel carries an almost equal share of the corresponding offered load. In this paper we study the problem of dynamic load balancing in broadcast WDM networks by retuning a subset of transceivers in response to changes in the overall traffic pattern. Assuming an existing wavelength assignment and some information regarding the new traffic demands, we present two approaches to obtaining a new wavelength assignment such that (a) the new traffic load is balanced across the channels, and (b) the number of transceivers that need to be retuned is minimized. The latter objective is motivated by the fact that tunable transceivers take a non-negligible amount of time to switch between wavelengths during which parts of the network are unavailable for normal operation. Furthermore, this variation in traffic is expected to take place over larger time scales (i.e., retuning will be a relatively infrequent event), making slowly tunable devices a cost effective solution. Our main contribution is a new approximation algorithm for the load balancing problem that provides for tradeoff selection, using a single parameter, between two conflicting goals, namely, the degree of load balancing and the number of transceivers that need to be retuned. This algorithm leads to a scalable approach to reconfiguring the network since, in addition to providing guarantees in terms of load balancing, the expected number of retunings scales with the number of channels, not the number of nodes in the network.

We consider a loss system model of interest in telecommunications. There is a single service facility with N servers and no waiting room. There are K types of customers, with type i customers requiring A_i servers simultaneously. Arrival processes are Poisson and service times are exponential. An arriving type i customer is accepted only if there are R_i \,(\geq A_i ) idle servers. We examine the asymptotic behavior of the above system in the regime known as critical loading where both N and the offered load are large and almost equal. We also assume that R_1,\ldots, R_{K-1} remain bounded, while R^N_K\to\infty and R^N_K/\sqrt{N}\to0 as N \to\infty. Our main result is that the K dimensional “queue length” process converges, under the appropriate normalization, to a particular K dimensional diffusion. We show that a related system with preemption has the same limit process. For the associated optimization problem where accepted customers pay, we show that our trunk reservation policy is asymptotically optimal when the parameters satisfy a certain relation.

In this paper, we develop approximation methods to analyze blocking in circuit switched networks with nonstationary call arrival traffic. We formulate generalizations of the pointwise stationary and modified offered load approximations used for the nonstationary Erlang loss model or M(t)/M/c/c queue. These approximations reduce the analysis of nonstationary circuit switched networks to solving a small set of simple differential equations and using the methods for computing the steady state distributions for the stationary versions of such loss networks. We also discuss how the use of time varying arrival rates literally adds a new dimension to the class of telecommunication networks we can model. For example, we can model the behavior of alternate routing due to link?failure, which is a feature that the classical stationary version of the model cannot capture. Our nonstationary model can also describe aspects of the dynamic calling traffic behavior arising in cellular mobile traffic. For the special case of a two?link, three node network, we present numerical results to compare the various approximation methods to calculations of the exact blocking probabilities. We also adapt these calculations to approximate the behavior of rerouting calling traffic due to link?failure. The results are achieved by formulating some new recursions for evaluating the steady state blocking probabilities of such networks. We also generalize these techniques to develop analogous formulas for a linear N?node circuit switched network.

In this paper we consider the M_t /G/\infty queueing model with infinitely many servers and a nonhomogeneous Poisson arrival process. Our goal is to obtain useful insights and formulas for nonstationary finite-server systems that commonly arise in practice. Here we are primarily concerned with the peak congestion. For the infinite-server model, we focus on the maximum value of the mean number of busy servers and the time lag between when this maximum occurs and the time that the maximum arrival rate occurs. We describe the asymptotic behavior of these quantities as the arrival changes more slowly, obtaining refinements of previous simple approximations. In addition to providing improved approximations, these refinements indicate when the simple approximations should perform well. We obtain an approximate time-dependent distribution for the number of customers in service in associated finite-server models by using the modified-offered-load (MOL) approximation, which is the finite-server steady-state distribution with the infinite-server mean serving as the offered load. We compare the value and lag in peak congestion predicted by the MOL approximation with exact values for M_t/M/s delay models with sinusoidal arrival-rate functions obtained by numerically solving the Chapman–Kolmogorov forward equations. The MOL approximation is remarkably accurate when the delay probability is suitably small. To treat systems with slowly varying arrival rates, we suggest focusing on the form of the arrival-rate function near its peak, in particular, on its second and third derivatives at the peak. We suggest estimating these derivatives from data by fitting a quadratic or cubic polynomial in a suitable interval about the peak.

An M/M/N queue, where each of the processors is subject to independent random breakdowns and repairs, is analyzed in the steady state under two limiting regimes. The first is the usual heavy traffic limit where the offered load approaches the available processing capacity. The (suitably normalized) queue size is shown to be asymptotically exponentially distributed and independent of the number of operative processors. The second limiting regime involves increasing the average lengths of the operative and inoperative periods, while keeping their ratio constant. Again the asymptotic distribution of an appropriately normalized queue size is determined. This time it turns out to have a rational Laplace transform with simple poles. In both cases, the relevant parameters are easily computable.

We consider a generalization of the classical Erlang loss model to multiple classes of customers. Each of the J customer classes has an associated Poisson arrival process and an arbitrary holding time distribution. Classj customers requireMj servers simultaneously. We determine the asymptotic form of the blocking probabilities for all customer classes in the regime known as critical loading, where both the number of servers and offered load are large and almost equal. Asymptotically, the blocking probability of classj customers is proportional toMj. This asymptotic result provides an approximation for the blocking probabilities which is reasonably accurate. We also consider the behavior of the Erlang fixed point (reduced load approximation) for this model under critical loading. Although the blocking probability of classj customers given by the Erlang fixed point is again asymptotically proportional toMj, the Erlang fixed point typically gives the wrong limit. Next we show that under critical loading the throughputs have a pleasingly simple form of monotonicity with respect to arrival rates: the throughput of classi is increasing in the arrival rate of classi and decreasing in the arrival rate of classj forj≠i. Finally, we compare two simple control policies for this system under critical loading.