PaperlayerView exact revision on arXiv

Source-linked AI summary

Introduction to Queueing Theory and Stochastic Teletraffic Models

Moshe Zukerman

arXiv:1307.2968v27math.PRcs.IT

TL;DR

The textbook addresses how stochastic models support telecommunications traffic modeling, resource provisioning, and traffic management. It develops queueing and teletraffic concepts through probability foundations, analytical results, and simulation-based learning. The book includes steady-state queueing results and applications such as Poisson departures from stable M/M/1 queues.

  • Problem

    The book addresses the need for foundational stochastic models applicable to telecommunications traffic modeling, resource provisioning, and traffic management.

  • Method

    The book combines probability and stochastic-process background, queueing theory, telecommunications applications, guided computer simulations, and analytical models.

  • Results

    Stable M/M/1 queues have a Poisson departure process with parameter λ, independent of the queue length after departures.

  • Takeaways & Limitations

    The material provides students with a foundation for understanding queueing behavior and telecommunications systems modeling through intuition, analysis, and simulation.

  • Takeaways & Limitations

    Strict-priority M/M/k/k blocking probabilities are not generally insensitive to holding-time distributions; insensitivity applies only to top-priority traffic.

Abstract

from arXiv · show

The aim of this textbook is to provide students with basic knowledge of stochastic models that may apply to telecommunications research areas, such as traffic modelling, resource provisioning and traffic management. These study areas are often collectively called teletraffic. This book assumes prior knowledge of a programming language, mathematics, probability and stochastic processes normally taught in an electrical engineering course. For students who have some but not sufficiently strong background in probability and stochastic processes, we provide, in the first few chapters, background on the relevant concepts in these areas.

16.2 Pollaczek-Khinchine Formula: by Kendall’s Recursion

This section builds the probability background used for stochastic modeling, covering events, random variables, distributions, conditional probability, independence, and joint distributions. It also introduces the exponential–geometric relationship and reports a capacity result.

  • Probability foundations: Probability theory provides the foundation for queueing theory and stochastic teletraffic models.
  • Events, Sample Space, and Random Variables: An experiment has a sample space of possible outcomes, while events are subsets that may be mutually exclusive or exhaustive.
  • Events, Sample Space, and Random Variables: A random variable is a real-valued function assigning a number to each possible experiment outcome, and functions or sums of random variables are also random variables.
  • Probability, Conditional Probability and Independence: Probability axioms include additivity for unions of mutually exclusive events, while relative frequencies provide an intuitive interpretation of probability.
  • Probability, Conditional Probability and Independence: Conditional probability restricts the sample space to a known event, and Bayes’ theorem reverses conditioning when the reverse conditional probability is easier to obtain.
  • Probability, Conditional Probability and Independence: Independence means that occurrence of one event does not affect the probability of the other.
  • Probability and Distribution Functions: A distribution function may refer to a probability, cumulative, or complementary distribution function, which are interconvertible.
  • Relationship between Exponential and Geometric Random Variables: As interval size δ and success probability p approach zero, the geometric and exponential distributions approach each other.

2 Relevant Background on Stochastic Processes

The section introduces stochastic processes as models of evolving random phenomena and develops classifications, stationarity, dependence measures, ergodicity, and point-process models. These concepts support later queueing and teletraffic analysis.

  • Stochastic-process foundations: Stochastic-process research combines theory, measurements, and modeling to match mathematical characteristics and dependencies to real processes.
  • Process definitions: A stochastic process is an indexed collection of random variables, with the index often representing time and the variable potentially counting telephone calls.
  • Process classifications: Processes are classified by whether time and state spaces are discrete or continuous, yielding four combinations.
  • Stationarity: Strict stationarity preserves joint probability functions under index shifts, whereas weak stationarity keeps the mean and covariance structure invariant over time.
  • Dependence measures: The autocovariance function measures covariance by lag, while the autocorrelation function is its normalized version; autocovariance sums and asymptotic variance rate are additional parameters.
  • Ergodicity: Ergodicity allows process realizations to contain enough information to estimate probabilistic properties such as the mean and autocovariance.
  • Point processes: Point processes with orderliness and memorylessness include discrete-time Bernoulli and continuous-time Poisson processes.
  • Markov chains: For an irreducible, aperiodic, and stable Markov chain, the long-run distribution forgets its initial distribution and steady-state probabilities follow steady-state equations.

3 General Queueing and Teletraffic Concepts

This section introduces queueing systems, their notation and performance measures, and basic steady-state relationships. It emphasizes simplifying complex telecommunications systems into analyzable models while retaining useful operational insight.

  • General Queueing and Teletraffic Concepts: Queueing systems are characterized by arrivals, service times, servers, buffer size, and queue discipline, but complex systems often require simpler models.
  • Notation: Kendall notation records the arrival process, service distribution, number of servers, buffer size, and queue discipline.
  • Notation: D, M, G, GI, and Geom identify common arrival or service assumptions, while examples such as D/D/1 and M/M/1 specify concrete queue models.
  • Notation: Because G/G/1 is used inconsistently, this book defines it as a single-server queue with generally distributed arrivals and service times without assuming independence.
  • Performance measures: Utilization is the average proportion of time servers are busy, and a stable G/G/1 queue satisfies ˆU = λ/µ when µ > λ.
  • Performance measures: For a G/G/1 queue, the empty-system probability can be obtained from utilization, while multi-server utilization averages individual server utilizations.
  • Performance measures: Little’s formula relates steady-state mean queue size and mean delay through λE[D] = E[Q].

4 Simulations

This section presents simulation as a way to estimate queueing performance when exact analysis is unavailable and to assess analytical approximations. It explains confidence intervals, simulation tradeoffs, and discrete-event simulation of G/G/1 queues.

  • Simulation rationale: Simulations estimate performance measures when analytical solutions are unavailable and evaluate the accuracy of analytical approximations.
  • Confidence intervals: Random-process simulations never produce exact steady-state measures, so repeated experiments are used to assess estimation error.
  • Confidence intervals: Student’s t-distribution provides confidence intervals for the sample mean, with examples using 95% confidence and n = 6 or n = 11 observations.
  • Confidence intervals: Larger observation counts generally produce smaller 95% confidence intervals, requiring a tradeoff between simulation time and accuracy.
  • Simulation exercises: Assignments vary sample sizes and distributions, including Pareto cases with 0 < γ ≤2 where the variance is infinite.
  • Discrete Event Simulation: Discrete-event simulation generates inter-arrival and service sequences, then records arrival, service, queue-size, and delay values for many G/G/1 customers.
  • Discrete Event Simulation: In the G/G/1 simulation table, service starts use the maximum of arrival and prior service-end times, service ends add duration, and delay subtracts arrival time.
  • Discrete Event Simulation: Queue-size-on-arrival observations do not generally give the steady-state queue-size distribution, which requires independent Poisson inspections.

5 Deterministic Queues

This section analyzes deterministic queues by tracking their cyclic or unbounded queue-size behavior and deriving utilization, mean queue size, and blocking probability. It covers single-server, multi-server, and finite-capacity systems under stable and unstable arrival conditions.

  • D/D/1: When λ > µ, D/D/1 is unstable: the queue grows without bound and utilization equals one.The server remains continuously busy because packets are always waiting.
  • D/D/1: For stable D/D/1, the queue alternates between zero and one, yielding utilization λ/µ.The cyclic process contains service for 1/µ during each cycle of length 1/λ.
  • D/D/1: The D/D/1 mean queue size equals its utilization and is no larger than that of any equivalent stable G/G/1 queue.Little’s formula gives the comparison because D/D/1 has zero waiting time, while other stable G/G/1 queues have E[WQ] ≥ 0.
  • D/D/k: For D/D/k, λ > kµ causes unbounded queue growth and utilization one, whereas λ ≤ kµ produces a cyclic process governed by the number of busy servers.The stable analysis identifies an integer ˆn satisfying (ˆn −1)µ < λ ≤ˆnµ and derives the mean queue size from the cycle.
  • D/D/k/k: In finite-capacity D/D/k/k, overload produces blocked arrivals; each cycle contains k + ˜n arrivals, of which ˜n are blocked.Blocking probability and mean queue size are obtained by analyzing the repeating cycle, while admitted customers enter service immediately.
  • Summary of Results: The summary table compares mean queue size and utilization for D/D/1, D/D/k, and D/D/k/k under load below or above capacity.At λ = kµ, the stable-case results apply under the convention that departures occur before simultaneous arrivals.

6 M/M/1

The M/M/1 results characterize queue-size, delay, utilization, busy-period, departure, and multiplexing behavior through the arrival-to-service ratio and spare capacity. They also provide dimensioning rules and extensions to infinite-server and general-service models.

  • Steady-state behavior: The stationary queue-size probabilities of a stable M/M/1 queue are geometric: π_i = ρ^i(1 −ρ), where ρ = λ/µ < 1.The result follows because the global and detailed balance equations are equivalent for M/M/1.
  • Departure process: A stable M/M/1 departure process is Poisson with parameter λ and independent of the post-departure queue length.This is the stated Burke’s theorem result for ρ < 1.
  • Busy periods and delays: The mean busy period equals the mean delay of a single customer in M/M/1, with the relationship explained using a preemptive LIFO formulation.The FIFO and LIFO queue-size processes and busy periods are statistically aligned, enabling the comparison.
  • Performance measures: For M/M/1, queue-size statistics depend only on ρ, whereas delay statistics depend on spare capacity µ −λ.Keeping ρ fixed while scaling λ and µ by N leaves the queue-size distribution unchanged but reduces mean delay to 1/[N(µ −λ)].
  • Multiplexing: Multiplexing recovers idle capacity: for N streams, the limiting gain approaches 1 −ρ, while FMUX can reduce mean delay N-fold and require less capacity than TDMA.For 100 users, FMUX requires 100.02 Mb/s versus 102 Mb/s for TDMA, saving 1.98 Mb/s.
  • Infinite-server models: For M/M/∞, the number of busy channels is Poisson, and the corresponding results extend to M/G/∞ because they depend only on mean service time.The mean number of busy servers is insensitive to the shape and higher moments of the service-time distribution.

8 M/M/k/k and Extensions

The M/M/k/k loss model uses Poisson arrivals and exponential, independent service times, with at most k simultaneous calls. Its Erlang B analysis characterizes blocking, carried and overflow traffic, and supports circuit dimensioning.

  • M/M/k/k has Poisson arrivals with parameter λ and independent exponential service times with parameter µ.
  • The carried traffic is (1−P_b)A, lower than the M/M/∞ carried traffic A because M/M/k/k has blocking.
  • The model is a truncated M/M/∞ system: its queue-size process cannot exceed k, and losses occur when state k is reached.
  • E_k(A) is the probability that all k circuits are busy and therefore the proportion of time new calls are blocked.
  • For A ≥ 10,000, choosing k = A gives no more than 1% blocking and no less than 99% utilization, according to the stated rule of thumb.
  • With exponential holding times, preemptive priority and non-priority systems can have the same blocking probability because both record one call loss in the described full-system arrival event.

2. Hayward Approximation.

The Hayward approximation and Equivalent Random Method estimate blocking in loss systems with non-Poisson traffic by replacing the traffic with an equivalent Erlang-B system. The methods differ in complexity, conservatism, and whether they estimate overflow variance.

  • The Equivalent Random Method models non-pure-chance traffic with mean M and variance V as overflow traffic from an equivalent Poisson primary system.
  • Equation (369) is an approximation, whereas Equation (370) is exact conditional on the approximation in Equation (369).
  • ERM can estimate both the mean and variance of secondary-system overflow traffic using the Riordan formula.
  • The Hayward approximation matches a mean-M, variance-V multi-server system to an M/M/k/k system with an adjusted offered load and rounded equivalent server count.
  • The Hayward approximation is simpler than ERM but cannot estimate overflow-traffic variance.
  • For M = 65 and V = 78, Hayward estimates 82 required servers, while ERM gives 82 with a less conservative choice and 83 with a conservative choice.
  • The comparison exercise evaluates Hayward, ERM, and a Poisson secondary-arrival approximation across a wide range of parameters.

9 M/M/k

The M/M/k queue generalizes M/M/1 to k exponential servers with an infinite buffer and Poisson arrivals. Its Erlang C analysis describes delay, queueing, utilization, and dimensioning under the stability condition A < k.

  • An M/M/k queue has Poisson arrivals of rate λ, k independent exponential servers of rate µ, and an infinite buffer.
  • The stability condition is λ < kµ, equivalently A = λ/µ < k.
  • Erlang C, C_k(A), is the proportion of time that all k servers are busy; when A/k > 1, the M/M/k queue is unstable.
  • When Q ≥ k, the queue evolves like an M/M/1 system with arrival rate λ and service rate kµ.
  • The mean queueing delay is the ratio between the probability that all servers are busy and the system’s spare capacity.
  • Dimensioning can find the smallest k satisfying a bound on Erlang C or mean delay, and utilization is the mean number of busy servers divided by k.
  • The chapter applies queueing theory to hospital resource allocation using an ICU example with exponentially distributed patient stays and Poisson arrivals.

10 Engset Loss Formula

The Engset loss model represents a finite customer population, making both arrival and service rates state-dependent. Its blocking probability differs from Erlang B because busy customers do not generate new calls.

  • 10 Engset Loss Formula: The Engset model applies to finite-source loss systems, unlike M/M/k/k models with Poisson traffic from an effectively infinite source population.
  • 10 Engset Loss Formula: With n busy circuits, the service rate is nµ, while the arrival rate depends on the number of free customers because busy customers do not originate calls.
  • 10.1 Steady-State Equations and Their Solution: The birth-and-death process has min{M,k}+1 states, reflecting the limits imposed by the source population and circuit count.
  • 10.2 Blocking Probability: When k ≥ M, the blocking probability is zero because the system cannot have more simultaneous calls than circuits.
  • 10.2 Blocking Probability: In Engset systems, π_k measures time congestion rather than call congestion, because the state-dependent arrival rate is lower when all circuits are busy.
  • 10.2 Blocking Probability: The Engset loss formula gives blocking probability for M > k, and interprets it as the time congestion of a system with M−1 sources.
  • 10.4 Insensitivity: The formula remains applicable to arbitrary idle-time and holding-time distributions, according to the stated insensitivity result.
  • 10.5 Load Classifications and Definitions: Reducing k with fixed M can increase offered traffic because more blocking leaves more idle customers available to attempt calls.

11 State Dependent SSQ

The section develops state-dependent single-server queue models in which arrival and service rates vary with queue size, yielding birth-and-death processes and steady-state solutions. It also distinguishes finite-capacity blocking behavior from buffer occupancy and shows that finite state spaces ensure stability.

  • Model formulation: State-dependent queues let arrival and service rates vary with the number of customers, supporting capacity increases or traffic throttling as queues grow.The model uses λ_i for arrivals and μ_i for service when i customers are present.
  • Model formulation: The queue-size process is represented by a birth-and-death Markov chain whose generator contains upward rates λ_i and downward rates μ_i.The steady-state equations are written as 0 = ΠQ, with local balance yielding recursive probabilities.
  • Steady-state analysis: Steady-state probabilities can be computed recursively as π_i = ρ_iπ_{i−1}, then normalized to obtain π_0 and all remaining π_i values.The recursive solution is contrasted with solving the full balance equations directly.
  • Blocking probability: Blocking probability is not generally equal to π_k when arrivals depend on state, because a full buffer may receive no arrivals and therefore cause no losses.The blocking probability is defined through arrivals occurring while the buffer is full; PASTA applies only when arrival rates are state-independent.
  • Finite-capacity behavior: Finite-capacity queues permit immediate service, waiting, or blocking, and extreme arrival patterns show that blocking probability cannot be inferred from average arrival rate alone.A single large burst can produce blocking probability approaching one even when the long-run arrival rate approaches zero.
  • Finite-capacity behavior: For M/M/1/N, the finite state space guarantees stability even when ρ > 1.This model is obtained as a special case of the state-dependent single-server queue.

13 Processor Sharing

The processor-sharing section analyzes queues in which server capacity is shared equally among customers, emphasizing service-size-dependent delays and insensitivity of several mean measures. It also identifies trade-offs with FIFO and a limitation concerning delay distributions.

  • M/M/1-PS model: Processor sharing gives every present customer an equal share of capacity and models time-shared processors and multiple TCP flows sharing a bottleneck.The M/M/1-PS model retains Poisson arrivals and exponential service requirements while changing the service discipline.
  • Delay by service requirement: Under processor sharing, a customer’s mean delay has a linear relationship with its required service time.A customer requiring twice as much service has twice the mean delay on average.
  • Comparison with FIFO: Processor sharing is better for small messages, whereas FIFO is better for large messages.The section presents this as a trade-off between the two service disciplines and motivates processor sharing when large jobs should not cause extreme delays for small ones.
  • Comparison with FIFO: M/M/1-PS and M/M/1-FIFO have the same steady-state queue-size probabilities and mean delay, but different delay distributions and variances.The PS delay variance exceeds FIFO’s by a factor that increases from 1 to 3 as ρ increases from 0 to 1.
  • Insensitivity: For M/G/1-PS, mean queue size, mean delay, and conditional mean delay are insensitive to the service-time distribution’s shape.The steady-state customer-count distribution is also insensitive, but the delay distribution is not.
  • Service-time variability: Lower message-size variance can yield lower mean queue size and delay than the corresponding exponential-service model.The text attributes this to fewer long messages delaying many small ones.

14 Multi-service Loss Model

The multi-service loss model represents classes whose customers require different numbers of servers, with class-specific blocking probabilities. Reversibility enables scalable exact solutions, while insensitivity makes blocking results depend only on mean holding times.

  • Model definition: Multi-service loss systems assign each customer class a server requirement and compute a class-specific blocking probability when insufficient servers are available.A class-i arrival is blocked if fewer than s_i free servers are available.
  • Reversibility and solution: Reversibility makes detailed balance equations sufficient for exact steady-state solutions and supports scalability to realistic system sizes.Detailed balance is described as easier to solve than global balance equations.
  • Insensitivity: Blocking probabilities depend on holding-time distributions only through their means, so the model remains exact for arbitrary and heavy-tailed holding times.The text specifically notes relevance to Internet flows with heavy-tailed distributions such as Pareto.
  • Applicability: The model addresses diverse bandwidth requirements and limited link capacity by evaluating blocking probabilities across service classes.The single-class M/M/k/k model is identified as a special case used in telephone-network design.
  • Example: In the two-class example, voice blocking probability is 0.425 and video blocking probability is 0.7.Voice requires one channel per call, whereas video requires two channels in a two-server system.
  • Critical loading: Under critical loading, asymptotic class-i blocking probability decays at rate 1/k and is linear in the server requirement s_i.A class requiring two servers therefore has twice the limiting blocking probability of a class requiring one server.

15 Discrete-Time Queue

The discrete-time queue models fixed-slot arrivals and one-cell service per slot using a Markov chain and its steady-state equations. A Z-transform derivation clarifies queue-size results and distinguishes slot-boundary emptiness from actual server idleness.

  • Discrete-time model: Discrete-time queues are useful when time is divided into fixed intervals and packets or cells have fixed lengths.The model serves one cell per slot whenever the queue is nonempty, with arrivals occurring at the beginning of the slot.
  • Markov-chain formulation: The queue-size process is a discrete-time Markov chain with a transition matrix and steady-state probability vector obtained from balance and normalization equations.The state space consists of all nonnegative integers.
  • Z-transform solution: The steady-state equations are transformed by multiplying by z^n and summing, producing an equation involving the Z-transforms Π(z) and A(z).The limit as z approaches 1 is then evaluated using L’Hopital’s rule.
  • Interpretation of π_0: The quantity π_0 is the probability that the queue is empty at a slot boundary, not the proportion of time the server is idle.Actual idle time is π_0a_0, incorporating the probability of no arrivals in the next slot.
  • Validation: The chapter exercises derive queue-size mean and variance with Z-transforms and verify them through simulations over varied parameter values.The verification is requested using confidence intervals.

16 M/G/1 and Extensions

This chapter extends queueing analysis from M/M/1 to M/G/1 systems with general service times, covering FIFO, LIFO, and priority disciplines. It derives mean delay and queue-size results and shows how service-time variability and scheduling discipline affect performance.

  • M/G/1 generalizes M/M/1 by allowing non-exponential IID service times, which removes the convenient Markov-chain structure and requires different analytical methods.The arrival process remains Poisson, with service times characterized by their mean and standard deviation.
  • The chapter analyzes M/G/1 queues under FIFO and LIFO service disciplines and with priorities.
  • The Pollaczek–Khinchine formulas give the mean delay and mean queue size for M/G/1 systems.The derivation uses residual service time and Little’s formula.
  • Higher service-time variance produces higher mean queue size and delay when arrival and service rates are equal.
  • 16.3 Special Cases: M/M/1 and M/D/1: M/M/1 has a mean queue size between one and two times that of equivalent M/D/1, with the ratio increasing toward 2 as utilization approaches 1.The difference is attributed to service variability in M/M/1 and is never greater than 100%.
  • The mean busy period is insensitive to service-time distribution shape when mean arrival and service rates are the same.
  • 16.5 M/G/1-LIFO: M/G/1-LIFO inherits M/M/1 results for mean queue size, mean delay, and queue-size distribution regardless of service-time distribution shape.
  • With stable priority queues, increasing service rate can make mean delay arbitrarily low, while priority remains beneficial when higher-priority utilization is below one but total utilization exceeds one.

17 Kingman’s Formula

The section develops queueing results for general input processes, including Kingman’s and Reich’s formulas, finite-buffer approximations, and network performance properties. It also illustrates these ideas through teletraffic throughput and resource-provisioning examples.

  • 17 Kingman’s Formula: Kingman’s formula approximates an infinite-buffer single-server queue under IID inputs and heavy traffic as ρ → 1.The inter-arrival and service-time variability enter through their coefficients of variation.
  • 17 Kingman’s Formula: The book asks students to verify Kingman’s consistency with the Pollaczek–Khinchine formula and assess its accuracy through simulation.The simulation exercise considers IID inter-arrivals and service times as utilization approaches one.
  • 18 Queues with General Input: General-input queues allow correlated inter-arrival times, which can produce blocking probabilities very different from those under IID or Poisson assumptions.The text contrasts ordinary M/M/1/N behavior with finite-buffer examples whose blocking probability approaches one at arbitrarily low utilization.
  • 18.1 Reich’s Formula: Reich’s formula is introduced alongside virtual waiting time, defined as the queueing delay excluding a packet’s own service.The corresponding remaining workload framework assumes neither particular inter-arrival times nor a specific service process.
  • 18.4 G/GI/1 Queue and Its G/GI/1/k Equivalent: For finite buffers, estimating unlimited-buffer overflow probability P(Q > k) is often easier than directly estimating blocking probability Ploss.The text emphasizes that blocking probability is the more practically important quantity because real buffers are finite.
  • 18.4 G/GI/1 Queue and Its G/GI/1/k Equivalent: A burst of 10^9 arrivals can yield Ploss ≈ 1 for a buffer of size one while the corresponding steady-state overflow probability is zero.After the burst is served, the infinite-buffer system remains empty, so P(Wq > k) = 0 in steady state.
  • 18.4 G/GI/1 Queue and Its G/GI/1/k Equivalent: Conversely, a queue with k = 10^9 can have Ploss ≈ 10^-9 while P(Wq > k) ≈ 1 under persistent arrivals and rare high service rates.This example demonstrates that finite-buffer loss and unlimited-buffer overflow need not track each other.
  • 20.1 Jackson Networks: Jackson’s theorem gives a product-form steady-state queue-size distribution and extends queue-size analysis to cyclic networks.The queues behave in queue-size statistics as if they were independent M/M/1, M/M/k, or M/M/∞ queues; a worked example requires μ ≥ 134.2 jobs per second for stability and QoS.

21 Stochastic Processes as Traffic Models

The section presents stochastic processes as traffic models by matching measurable traffic characteristics and selecting models suited to queueing analysis. It discusses Poisson, MMPP, Gaussian autoregressive, and exponential autoregressive models, along with their assumptions and limitations.

  • Traffic modeling seeks stochastic processes that characterize network traffic load for performance estimation and resource provisioning.
  • Models may be fitted to real traffic using characteristics such as mean, variance, and autocorrelation-related measures.For long-range-dependent processes, time-dependent IDC or the Hurst parameter may be more appropriate than AVR when the latter is not finite.
  • The Poisson process is analytically convenient and useful for traffic from many independent sparse sources, but it is inaccurate for packet streams from few users.Its single parameter and memoryless nature support telephone-network design, while practitioners may still use it for packet traffic despite the stated inaccuracy.
  • MMPP models capture bursty traffic statistics better than Poisson models while remaining amenable to queueing analysis.The simplest MMPP(2) has four parameters: λ0, λ1, δ0, and δ1.
  • Gaussian autoregressive models represent traffic through autoregressive dependence and Gaussian innovations, with AR(1) used to reduce parameter-fitting complexity.The AR(1) model was proposed for a variable-bit-rate traffic stream generated by a single video-telephony source.
  • Traffic-model fidelity is evaluated through queueing performance rather than requiring every modeled traffic value to match real traffic.The Gaussian autoregressive process can produce negative values, but the model is retained when its queueing behavior is sufficiently close to that driven by real traffic.
  • EAR(1) produces exponential inter-arrival times, which become independent only when a = 0, corresponding to a Poisson process.The process is defined using IID Bernoulli variables and IID exponential variables with parameter λ.
Loading 1307.2968v27…