Network Mathematics, Traffic and Queueing Models

Introduction and Context

Communication networks are systems in which users, applications, and devices compete for shared resources such as links, buffers, processors, radio channels, and switching fabrics. The performance of such systems cannot be assessed only by nominal data rate. A network may have adequate bandwidth but still exhibit excessive delay, packet loss, or unstable queue growth when traffic is bursty or when service capacity is poorly allocated.

Network mathematics provides the analytical language for these questions. Probability describes uncertainty in arrivals, service times, failures, and user behavior. Stochastic processes represent the time evolution of traffic and network state. Queueing theory relates arrival rates and service rates to waiting time, buffer occupancy, utilization, throughput, and loss. These models support capacity planning, protocol design, traffic engineering, dimensioning of wireless and wired systems, and performance assurance in cloud and edge networks.

The models in this chapter are idealizations. Their value lies not in reproducing every implementation detail, but in identifying dominant mechanisms and providing transparent estimates. Sound engineering practice therefore combines analytical models with measurements, simulation, and sensitivity analysis.

Learning Objectives

After completing this chapter, the reader should be able to:

1. Distinguish random variables, stochastic processes, rates, probabilities, and distributions used in network analysis.

2. Characterize traffic using mean rate, variance, burstiness, correlation, and offered load.

3. Apply Little’s law and interpret its assumptions.

4. Explain discrete-time and continuous-time Markov chains and derive stationary probabilities for simple systems.

5. Use Poisson and exponential models appropriately, while recognizing their limitations.

6. Analyze M/M/1, M/M/c, M/M/1/K, and loss systems.

7. Estimate utilization, delay, throughput, blocking probability, loss, and capacity requirements.

8. Evaluate analytical assumptions and select suitable models for practical network design.

1. Mathematical Foundations for Network Analysis

1.1 Random variables and distributions

A random variable assigns a numerical value to an uncertain outcome. In networking, examples include the number of packets arriving during an interval, packet length, interarrival time, service time, round-trip delay, or number of active users.

A discrete random variable has a probability mass function (PMF), p_X(x) = P(X = x). A continuous random variable has a probability density function (PDF), f_X(x), and probabilities are obtained by integration. The cumulative distribution function is

F_X(x) = P(X ≤ x).

The expected value and variance are, respectively,

E[X] = Σ_x x p_X(x) for a discrete variable,

E[X] = ∫ x f_X(x) dx for a continuous variable,

Var(X) = E[(X − E[X])²].

The standard deviation is the square root of the variance. The coefficient of variation is

C_X = standard deviation of X / E[X],

provided E[X] is nonzero. It is a useful dimensionless measure of variability. A deterministic quantity has C_X = 0; an exponential random variable has C_X = 1.

For a random variable representing packet arrivals in a time interval, the mean describes average demand, whereas variance describes fluctuation around that demand. Two traffic sources can have the same mean rate but very different queueing behavior if one is smooth and the other is bursty.

1.2 Rates, units, and offered load

An arrival rate λ is measured in events per unit time, such as packets/s, calls/minute, or flows/s. A service rate μ is measured in services per unit time. The mean service time is 1/μ when service times are exponentially distributed or when μ is defined as the reciprocal of the mean service time.

For a single server, the offered traffic intensity or utilization is commonly written as

ρ = λ/μ = λ E[S],

where S is service time. Since λ has units of events/time and E[S] has units of time/event, ρ is dimensionless. For stable operation of an infinite-buffer single-server queue, ρ must be less than 1. In a multi-server system with c identical servers,

ρ = λ/(cμ).

The offered load in teletraffic is often expressed in Erlangs:

A = λ E[H],

where λ is the call arrival rate and H is the mean holding time. One Erlang represents one continuously occupied resource on average. Erlangs are dimensionless, although the term is retained to identify traffic intensity.

1.3 Independence and conditional probability

Many tractable models assume that arrivals and service times are independent, and that successive interarrival or service times are identically distributed. Independence means that knowledge of one variable does not change the probability distribution of another. In practice, network traffic often exhibits dependence caused by sessions, feedback, retransmissions, diurnal patterns, and correlated user activity.

Conditional probability is defined by

P(A|B) = P(A ∩ B)/P(B),

for P(B) > 0. Bayes’ rule follows as

P(A|B) = P(B|A)P(A)/P(B).

These concepts are useful in fault diagnosis, admission control, anomaly detection, and Bayesian estimation of traffic states.

2. Traffic Characterization

2.1 Packet, flow, and session viewpoints

Traffic can be measured at several levels. Packet-level analysis studies individual arrivals and service times. Flow-level analysis groups packets belonging to a transport or application flow. Session-level analysis considers user activities such as web transactions, calls, video sessions, or industrial control episodes. A model appropriate at one level may be misleading at another.

A network administrator should specify the observation interval and aggregation level before estimating a rate. If N(T) packets arrive during interval T, the measured average rate is

λ̂ = N(T)/T.

As T increases, short-term fluctuations are smoothed. However, long intervals may conceal bursts that determine buffer requirements.

2.2 Burstiness and variability

A convenient first indicator of burstiness is the ratio of peak or percentile rate to mean rate. More formal measures include variance-to-mean ratios, coefficient of variation, autocorrelation, and the Hurst parameter for long-range-dependent traffic.

The autocorrelation function of a stationary process X(t) describes dependence between observations separated by lag τ. Significant positive correlation means that high arrivals tend to be followed by high arrivals, increasing the likelihood of sustained queue buildup. Classical Markovian queues generally assume independent arrivals and therefore may underestimate delay under correlated traffic.

Self-similar traffic exhibits similar burst structure across time scales. Internet traffic can show such behavior because of aggregated file transfers, video sessions, and application-level synchronization. In these cases, a short-memory Poisson model may be useful for a baseline but insufficient for buffer dimensioning.

2.3 Stationarity and nonstationarity

A stationary traffic process has statistical properties that do not change with time, at least over the modeling interval. Real networks are often nonstationary: working-hour traffic, flash crowds, mobility, outages, and software updates produce time-varying arrival rates λ(t).

A practical approach is piecewise-stationary modeling. The day is divided into intervals for which the rate is approximately constant, and a separate queueing estimate is obtained for each interval. When λ(t) changes rapidly relative to the queue’s response time, transient analysis or discrete-event simulation is preferable.

2.4 Capacity and service time

For a link with capacity C bits/s and mean packet length L bits, the approximate packet service rate is

μ ≈ C/L,

assuming one server, negligible protocol overhead, and a constant or representative packet length. If packet lengths vary, the service time is S = L/C and E[S] = E[L]/C. Variability in L creates variability in service time, which can increase waiting compared with a fixed-length approximation.

Capacity planning must distinguish link rate from usable service rate. Framing overhead, coding, contention, processing, retransmissions, and reserved capacity reduce the effective rate. The analytical value of C should therefore be the rate actually available to the modeled traffic class.

3. Little’s Law

Little’s law is one of the most broadly applicable results in performance analysis:

L = λW.

Here L is the long-run time-average number of items in a system, λ is the long-run average throughput, and W is the average time an item spends in the system. The system may be a queue, a router, a link, a complete network, or a workflow. For a queue alone,

L_q = λ W_q.

The law applies under stable long-run operation and suitable flow conservation: items entering and leaving must be counted consistently, and averages must exist. It does not require Poisson arrivals, exponential service, or a particular queue discipline.

If a router carries 12,000 packets/s and the measured average number of packets in the router—including packets waiting and in service—is 240, then

W = L/λ = 240/12,000 = 0.02 s = 20 ms.

If the average service time is 2 ms, the average queueing delay is approximately 18 ms. This decomposition assumes that the system time includes service time and that the measurements refer to the same population.

Little’s law is especially valuable for checking measurements. If independently measured throughput, occupancy, and delay do not satisfy L ≈ λW, then counters may use different scopes, the system may be transient, or traffic may be lost or duplicated between measurement points.

4. Markov Chains and Stochastic Processes

4.1 Discrete-time Markov chains

A discrete-time Markov chain consists of states X_n and transition probabilities

P_ij = P(X_{n+1} = j | X_n = i).

The Markov property states that the next state depends on the present state, not on the complete past. The transition matrix P has nonnegative entries and each row sums to one.

A stationary distribution π satisfies

π = πP, Σ_i π_i = 1.

For an irreducible, positive recurrent finite-state chain, the stationary distribution exists and is unique. It describes long-run state probabilities. For example, states may represent the number of packets in a finite buffer, channel conditions, or the operational status of a network component.

4.2 Continuous-time Markov chains

In a continuous-time Markov chain (CTMC), transitions occur at random times. If the holding time in a state is exponentially distributed and transition rates are fixed, the process has the continuous-time Markov property. Queue length is a common CTMC state variable in M/M/1 and related models.

For a birth–death process, transitions from state n to n+1 occur at birth rate λ_n, and transitions from n to n−1 occur at death rate μ_n. In equilibrium, local balance often gives

π_n λ_n = π_{n+1} μ_{n+1}.

This relation is a powerful route to stationary probabilities, but it must be used with correct state-dependent rates and valid equilibrium assumptions.

4.3 Markov model limitations

Markov models are attractive because they are mathematically tractable and can support online estimation. Their memoryless assumption is not universally realistic. If service times are deterministic, heavy-tailed, or strongly correlated, an exponential model may produce inaccurate delay or tail estimates. Phase-type distributions, semi-Markov models, Markov-modulated Poisson processes, and simulation can represent richer behavior.

5. Poisson Arrivals and Exponential Service

A Poisson process with rate λ has independent increments, and the number of arrivals N(t) in an interval of length t has distribution

P[N(t)=n] = e^(−λt)(λt)^n/n!, n = 0,1,2,...

Its mean and variance are both λt. The interarrival time is exponentially distributed with mean 1/λ:

f_A(a) = λe^(−λa), a ≥ 0.

The exponential distribution is memoryless:

P(A>s+t | A>s) = P(A>t).

A Poisson model is often a reasonable approximation when many independent users contribute relatively small traffic and events are not synchronized. It is inappropriate when arrivals are scheduled, strongly correlated, admission-controlled, or dominated by a small number of bursty sources.

The notation M/M/1 means Markovian (exponential) interarrival times, Markovian service times, and one server. More generally, Kendall’s notation is commonly written A/S/c/K/N/D, where A denotes arrival distribution, S service distribution, c number of servers, K system capacity, N source population, and D service discipline. Omitted descriptors are conventionally treated according to the standard notation, but assumptions should always be stated explicitly.

6. The M/M/1 Queue

Consider a single server with Poisson arrival rate λ, exponential service rate μ, unlimited waiting room, first-come-first-served service, and ρ = λ/μ < 1.

The stationary probability of n items in the system is

P_n = (1−ρ)ρ^n, n ≥ 0.

Thus the probability that the server is busy is ρ, and the probability of an empty system is 1−ρ.

The mean number in the system is

L = ρ/(1−ρ).

The mean number waiting in the queue is

L_q = ρ²/(1−ρ).

Using Little’s law, the mean system time and queueing time are

W = L/λ = 1/(μ−λ),

W_q = L_q/λ = ρ/(μ−λ) = λ/[μ(μ−λ)].

The relation W = W_q + 1/μ also holds. As ρ approaches 1, both delay and queue occupancy increase without bound. This nonlinear behavior explains why operating a link continuously near its nominal capacity can create unacceptable latency.

Worked Example 1: Single router output queue

A router output port serves packets at μ = 1,000 packets/s. Packets arrive according to a Poisson process at λ = 800 packets/s. Then

ρ = 800/1,000 = 0.8.

The queue is stable because ρ < 1. The mean number in the system is

L = 0.8/(1−0.8) = 4 packets.

The mean system delay is

W = 1/(1,000−800) = 0.005 s = 5 ms.

The mean queueing delay is

W_q = W − 1/μ = 5 ms − 1 ms = 4 ms.

The mean number waiting is λW_q = 800 × 0.004 = 3.2 packets. The difference between 4 and 3.2 is the average one packet in service, weighted by server utilization: ρ = 0.8.

If arrivals increase to 950 packets/s while service remains 1,000 packets/s, W becomes 20 ms. A 15.6% increase in arrival rate from 800 to 950 packets/s produces a fourfold increase in mean system delay, illustrating the danger of high utilization.

7. Multiple Servers and Finite Capacity

7.1 M/M/c queues

An M/M/c queue has c identical parallel servers, Poisson arrivals, exponential service at rate μ per server, unlimited waiting space, and a common queue. Stability requires λ < cμ. Define a = λ/μ and ρ = λ/(cμ).

The probability that the system is empty is

P_0 = [Σ_{n=0}^{c−1} a^n/n! + a^c/(c!(1−ρ))]^(−1).

The Erlang C probability that an arrival must wait is

P_wait = [a^c/(c!(1−ρ))]P_0.

The mean queue length is

L_q = P_wait · ρ/(1−ρ),

and W_q = L_q/λ. The mean system time is W = W_q + 1/μ. These formulas assume that waiting room is unlimited and that arrivals are not lost.

7.2 Finite-buffer M/M/1/K

In an M/M/1/K system, at most K items can be in the system, including the item in service. Arrivals finding K items are blocked or lost. For ρ ≠ 1,

P_n = [(1−ρ)/(1−ρ^(K+1))]ρ^n, 0 ≤ n ≤ K.

The blocking probability is

P_block = P_K.

The effective throughput is

λ_eff = λ(1−P_block).

The server utilization is λ_eff/μ, not necessarily λ/μ, because blocked arrivals receive no service. Mean delay for admitted traffic must be calculated using admitted throughput and the mean number of items in the system, not blindly by substituting the offered arrival rate into Little’s law.

7.3 Loss systems and Erlang B

An Erlang loss system has c servers and no waiting room. An arriving call that finds all servers occupied is blocked and lost. With offered traffic A in Erlangs, the blocking probability is

B(c,A) = (A^c/c!) / Σ_{k=0}^{c} (A^k/k!).

This is the Erlang B formula. It assumes Poisson arrivals, exponentially distributed holding times, independent servers, and immediate loss when all servers are busy. It is central to trunk dimensioning, wireless channel allocation, and circuit-switched systems.

Worked Example 2: Wireless channel dimensioning

A wireless access system receives 18 call attempts per hour, and the mean holding time is 3 minutes = 0.05 hour. The offered traffic is

A = 18 × 0.05 = 0.9 Erlang.

Suppose two channels are available. The blocking probability is

B(2,0.9) = (0.9²/2) / [1 + 0.9 + 0.9²/2]

= 0.405/2.305

≈ 0.1757.

Thus approximately 17.6% of calls are blocked, which may be unacceptable. With three channels,

B(3,0.9) = (0.9³/6) / [1 + 0.9 + 0.405 + 0.1215]

= 0.1215/2.4265

≈ 0.0501.

Three channels reduce blocking to approximately 5.0%. If the target is 1%, further capacity is required; the calculation should be repeated for four or more channels rather than relying on utilization alone.

8. Throughput, Delay, Loss, and Capacity Decisions

Performance metrics must be defined precisely:

• Offered load is traffic presented to the system.
• Admitted load is traffic accepted for service.
• Throughput is traffic successfully completed or transmitted.
• Utilization is the fraction of service capacity occupied.
• Queueing delay is time waiting before service.
• System delay includes waiting and service.
• Blocking occurs when an arrival is denied admission.
• Loss occurs when traffic is discarded, dropped, or fails to complete.

In a lossless stable single-server system, throughput equals the external arrival rate. In a finite or lossy system, throughput is lower than offered arrival rate. Confusing these quantities can produce serious capacity errors.

Capacity should be selected using a performance target, not merely the condition ρ < 1. For example, a designer may require mean delay below 10 ms, 99th-percentile delay below 50 ms, blocking below 1%, or utilization below 70% during the busy hour. Mean-value models do not directly guarantee percentile objectives. Tail requirements often require distributional analysis, large-deviation methods, measurement, or simulation.

A useful engineering workflow is:

1. Define the traffic population, measurement point, and performance objective.

2. Estimate arrival and service distributions, including variability and time dependence.

3. Select the simplest model whose assumptions are defensible.

4. Compute utilization, delay, occupancy, throughput, and loss.

5. Perform sensitivity analysis for burstiness, demand growth, and service degradation.

6. Validate against measurements or discrete-event simulation.

7. Add operational margins and monitor the system after deployment.

9. Applied Case: Illustrative Edge-Computing Gateway

Background

Consider an illustrative edge-computing gateway serving industrial sensors. Each sensor submits short inference requests to a processor pool. During a production shift, the gateway receives an estimated 720 requests per second. Profiling indicates that one processor can complete an average of 300 requests per second. The initial design uses three identical processors and a common queue.

Problem

The organization requires stable operation and seeks to limit average queueing delay. It also wants to understand the effect of a 20% demand increase caused by additional sensors.

Analysis

For three processors,

cμ = 3 × 300 = 900 requests/s,

ρ = 720/900 = 0.8.

The system is stable under an M/M/3 approximation, but utilization is high. Let a = λ/μ = 720/300 = 2.4. The empty-system probability is

P_0 = [1 + 2.4 + 2.4²/2! + 2.4³/(3!(1−0.8))]^(−1).

The terms are 1, 2.4, 2.88, and 2.304/(0.2) = 11.52. Therefore,

P_0 = 1/17.8 ≈ 0.0562.

The probability an arrival waits is

P_wait = [2.4³/(6(1−0.8))]P_0

= 11.52 × 0.0562

≈ 0.647.

The mean queue length is

L_q = 0.647 × 0.8/0.2 ≈ 2.588 requests.

Thus

W_q = L_q/λ ≈ 2.588/720 = 0.00359 s = 3.59 ms.

The mean service time is 1/300 = 3.33 ms, so the mean system time is approximately 6.92 ms.

With a 20% demand increase, λ becomes 864 requests/s and ρ = 864/900 = 0.96. The system remains mathematically stable, but the waiting probability and delay rise sharply. This is an operationally fragile design: a processor slowdown, traffic burst, or estimation error can cause substantial queue growth.

Approach and solution

The engineering team evaluates a fourth processor. At λ = 864 requests/s and c = 4, total service capacity is 1,200 requests/s and ρ = 0.72. Recomputing the M/M/c expressions gives substantially lower waiting probability and delay. The team also introduces admission control for noncritical inference jobs, measures service-time variability, and monitors queue percentiles rather than only averages.

Outcome

The design decision is not based solely on current mean load. It incorporates growth, uncertainty, service degradation, and application-level priority. The additional processor costs capacity but provides resilience and a practical latency margin.

Learning points

1. Stability is necessary but not sufficient for acceptable performance.

2. Small increases near high utilization can cause disproportionate delay increases.

3. Multi-server formulas help quantify the value of parallel capacity.

4. Analytical results should be supplemented by measurements when service times and arrivals are not exponential.

5. Capacity decisions should include growth and failure margins.

10. Limitations, Governance, and Contemporary Practice

Queueing formulas can mislead when traffic is nonstationary, service times are heavy-tailed, buffers are managed by complex scheduling, or packets interact through retransmission and congestion control. Priority queues may give one class low delay while increasing delay for another. Wireless systems add fading, interference, mobility, and correlated channel availability. Cloud systems add autoscaling delays, placement constraints, and shared-resource interference.

Model risk should be governed explicitly. Analysts should document data sources, observation intervals, omitted traffic classes, assumptions, and uncertainty ranges. Capacity recommendations should not be presented as exact guarantees when the model is only an approximation. Privacy-preserving aggregation is important when traffic measurements can reveal user behavior, industrial operations, or sensitive communications.

Modern networks increasingly combine analytical models with telemetry, simulation, and machine learning. AI can forecast time-varying demand, detect anomalous queue behavior, and optimize admission or routing. However, predictive models may fail under distribution shift, and automated controls can amplify congestion if feedback delays are ignored. Explainability, safeguards, rollback procedures, and human oversight are therefore important in operational deployments.

Best practice is to use queueing models as interpretable baselines, validate them against production data, recalibrate them as traffic changes, and report both central estimates and uncertainty. When tail latency, correlated bursts, or transient overload dominate the objective, simulation or richer stochastic models should replace simplistic steady-state formulas.

Chapter Summary

Network performance analysis begins with precise definitions of traffic, service, capacity, throughput, delay, and loss. Probability and stochastic processes describe uncertainty and time evolution, while Poisson and exponential assumptions provide tractable Markovian models. Little’s law, L = λW, connects average occupancy, throughput, and time under stable flow conditions and is applicable far beyond a particular queue type.

The M/M/1 model shows how utilization controls delay and queue length, with instability when λ ≥ μ. M/M/c models quantify the benefit of parallel servers; finite-buffer queues account for blocking and loss; and Erlang B supports dimensioning of systems with no waiting room. Offered load and admitted throughput must be distinguished in lossy systems. High utilization may appear efficient but can create severe delay sensitivity and weak resilience.

These models are foundations rather than universal descriptions. Traffic burstiness, correlation, nonstationarity, variable packet sizes, priorities, retransmissions, and tail objectives may require richer models, simulation, or measurement. Professional capacity planning therefore combines mathematical analysis, sensitivity testing, empirical validation, operational margins, and responsible governance.

Key Terms

Random variable: A numerical representation of an uncertain outcome.

Expected value: The probability-weighted long-run average of a random variable.

Variance: A measure of squared deviation from the mean.

Coefficient of variation: Standard deviation divided by the mean.

Arrival rate: Average number of arrivals per unit time.

Service rate: Average number of completed services per unit time per server.

Utilization: Fraction of service capacity occupied.

Offered load: Traffic presented to a system before blocking or loss.

Erlang: Dimensionless unit of offered teletraffic intensity.

Stationarity: Condition in which relevant statistical properties remain constant over time.

Poisson process: Counting process with independent increments and Poisson interval counts.

Exponential distribution: Memoryless distribution commonly used for interarrival or service times.

Markov chain: Stochastic process whose next state depends on the current state.

Little’s law: Relationship L = λW between mean population, throughput, and mean time.

Queueing delay: Time spent waiting before service begins.

System time: Total time in a system, including waiting and service.

Blocking probability: Probability that an arrival is denied access because capacity is unavailable.

M/M/1: Single-server queue with Poisson arrivals and exponential service.

Erlang B: Formula for blocking in a multi-server loss system with no waiting room.

Burstiness: Degree to which traffic arrives unevenly or in concentrated episodes.

Review Questions

Short Answer Questions

1. Define utilization for a single-server queue and state the stability condition for an infinite-buffer M/M/1 system.

2. What is the difference between offered load and throughput in a lossy network?

3. State Little’s law and identify its three principal quantities.

4. Why can two traffic sources with the same mean rate produce different queueing delays?

5. What does the memoryless property of the exponential distribution mean?

Long Answer Questions

1. Explain the assumptions, stationary probabilities, and performance measures of the M/M/1 queue.

2. Derive the relationship between mean queue length and mean waiting time using Little’s law, and discuss the required conditions.

3. Compare M/M/1, M/M/c, finite-buffer M/M/1/K, and Erlang loss models in terms of servers, waiting room, blocking, and applicability.

4. Discuss how burstiness, autocorrelation, and nonstationarity affect the validity of Poisson queueing models.

5. Develop a capacity-planning methodology for a network service with delay and blocking constraints.

Analytical / Application Questions

1. A link serves 500 packets/s and receives Poisson traffic at 350 packets/s. Under an M/M/1 model, calculate utilization, mean number in the system, mean system time, and mean queueing time.

2. A service desk has four identical servers, arrival rate 10 jobs/minute, and service rate 3 jobs/minute per server. Determine utilization and use the M/M/c framework to estimate the probability that an arriving job must wait.

3. A finite-buffer router has capacity for K packets including the packet in service. Explain how increasing K affects blocking and delay, and identify circumstances in which increasing the buffer may worsen application performance.

Multiple Choice Questions with Answers

Question 1

For a stable M/M/1 queue, which condition must hold?

A. λ = 0

B. λ < μ

C. λ > μ

D. ρ > 1

Correct Answer: B. λ < μ

Question 2

Little’s law is correctly written as:

A. L = W/λ

B. L = λW

C. W = Lλ

D. λ = LW

Correct Answer: B. L = λW

Question 3

If λ = 800 packets/s and μ = 1,000 packets/s in an M/M/1 queue, utilization is:

A. 0.08

B. 0.20

C. 0.80

D. 1.25

Correct Answer: C. 0.80

Question 4

For an M/M/1 queue with λ = 800 packets/s and μ = 1,000 packets/s, the mean system time is:

A. 1 ms

B. 4 ms

C. 5 ms

D. 20 ms

Correct Answer: C. 5 ms

Question 5

In a Poisson process with rate λ, the number of arrivals in time t has which distribution?

A. Binomial with mean t/λ

B. Poisson with mean λt

C. Exponential with mean λt

D. Normal with variance λ/t

Correct Answer: B. Poisson with mean λt

Question 6

An Erlang of offered traffic represents:

A. One bit per second

B. One continuously occupied resource on average

C. One blocked call per hour

D. One server with deterministic service

Correct Answer: B. One continuously occupied resource on average

Question 7

In an M/M/1/K queue, an arrival finding K items in the system is generally:

A. Served twice

B. Placed in an infinite external queue

C. Blocked or lost

D. Converted into a Poisson arrival

Correct Answer: C. Blocked or lost

Question 8

A call system receives 20 calls per hour with mean holding time 0.1 hour. Its offered traffic is:

A. 0.2 Erlang

B. 2 Erlangs

C. 20 Erlangs

D. 200 Erlangs

Correct Answer: B. 2 Erlangs

Question 9

Which feature most directly violates the independent-increment assumption of a simple Poisson model?

A. Many independent users

B. Constant average rate

C. Strong synchronization of arrivals

D. A long observation interval

Correct Answer: C. Strong synchronization of arrivals

Question 10

As utilization of an M/M/1 queue approaches one from below, mean system delay:

A. Approaches zero

B. Remains constant

C. Increases without bound

D. Equals the service time exactly

Correct Answer: C. Increases without bound