Queueing Systems, Volume 1¶
Kleinrock, L. (1975). Queueing Systems, Volume 1: Theory. Wiley.
Cited by¶
13 citations across 13 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Backpressure
- The same algebra is queueing theory's stability condition \(\rho = \lambda/\mu < 1\): when the arrival rate \(\lambda\) (producer) exceeds the service rate \(\mu\) (consumer), queue length grows without bound, so a stable system must hold \(\lambda \le \mu\) — which an explicit backpressure channel enforces by clamping \(\lambda\) to the consumer's \(\mu\).
This sourceEstablishes the stability condition ρ = λ/μ < 1, under which queue length stays bounded only when arrival rate does not exceed service rate.
- The same algebra is queueing theory's stability condition \(\rho = \lambda/\mu < 1\): when the arrival rate \(\lambda\) (producer) exceeds the service rate \(\mu\) (consumer), queue length grows without bound, so a stable system must hold \(\lambda \le \mu\) — which an explicit backpressure channel enforces by clamping \(\lambda\) to the consumer's \(\mu\).
- Buffering
- Queueing theory model: A single-server queue M/M/1 with arrival rate λ and service rate μ receives jobs according to a Poisson process, the canonical model Kleinrock (1975) develops in detail.
This sourceStandard queueing-theory reference: develops the M/M/1 model (Poisson arrivals, exponential service, single server), deriving steady-state buffer occupancy ρ/(1−ρ) and characterizing stability, blocking, and delay.
- Queueing theory model: A single-server queue M/M/1 with arrival rate λ and service rate μ receives jobs according to a Poisson process, the canonical model Kleinrock (1975) develops in detail.
- Client Server Model
- Recognizing a system as client–server makes four standard structural moves available. Capacity reasoning: because the server is a shared resource, the apparatus of queueing theory, load shedding, admission control, and fairness across clients applies directly — a clinic with one specialist serving many patients has the same capacity equation as a server with one processing unit serving many connections.
This sourceFoundational treatment of queueing theory — admission control, load, and fairness across many clients contending for one shared server.
- Recognizing a system as client–server makes four standard structural moves available. Capacity reasoning: because the server is a shared resource, the apparatus of queueing theory, load shedding, admission control, and fairness across clients applies directly — a clinic with one specialist serving many patients has the same capacity equation as a server with one processing unit serving many connections.
- Interference and Contention
- Contention is not inherently a design flaw—many systems intentionally multiplex shared resources—but it produces measurable degradation: increased latency, reduced throughput, dropped transactions, or quality loss, as Kleinrock (1975) developed quantitatively in queueing-theoretic terms.
This sourceStandard queueing-theory reference: develops the M/M/1 model (Poisson arrivals, exponential service, single server), deriving steady-state buffer occupancy ρ/(1−ρ) and characterizing stability, blocking, and delay distributions.
- Contention is not inherently a design flaw—many systems intentionally multiplex shared resources—but it produces measurable degradation: increased latency, reduced throughput, dropped transactions, or quality loss, as Kleinrock (1975) developed quantitatively in queueing-theoretic terms.
- Logistic Growth
- And in software systems, queue lengths, cache fill, and throughput-versus-load curves show sigmoidal saturation as capacity is consumed.
This sourceStandard reference for the theory of queues, deriving how queue lengths and delay grow without bound (and throughput saturates) as offered load approaches finite system capacity — the sigmoidal load-versus-throughput behavior of software systems. ISBN 978-0-471-49110-1.
- And in software systems, queue lengths, cache fill, and throughput-versus-load curves show sigmoidal saturation as capacity is consumed.
- Markov Process
- Birth–death models of population dynamics and many models of molecular evolution share this structure. Queueing / operations: Birth–death and M/M/1 models assume memoryless arrivals and service (exponential inter-event times), making the present queue length a sufficient state for the future and rendering the system analytically tractable.
This sourceStandard queueing-theory reference: develops the M/M/1 model (Poisson arrivals, exponential service, single server), deriving steady-state buffer occupancy ρ/(1−ρ) and characterizing stability, blocking, and delay distributions.
- Birth–death models of population dynamics and many models of molecular evolution share this structure. Queueing / operations: Birth–death and M/M/1 models assume memoryless arrivals and service (exponential inter-event times), making the present queue length a sufficient state for the future and rendering the system analytically tractable.
- Queueing
- Not free of instability near saturation:
This sourceStandard reference developing the M/M/1 model and showing wait time W = 1/(μ−λ) diverges as utilization → 1, characterizing instability near saturation.
- Not free of instability near saturation:
- Tempo Mismatch
- Thundering Herd
- Standard queueing results (e.g., the M/M/1 model) assume the arrivals are independent, giving bounded expected waiting time.
This sourceStandard reference for queueing theory, including the M/M/1 model and the independent-arrivals (Poisson) assumption that a correlated, synchronized release violates.
- Standard queueing results (e.g., the M/M/1 model) assume the arrivals are independent, giving bounded expected waiting time.
- Unevenness Waste
- The nonlinear-interaction role is precisely this term: at \(\rho = 0.5\) the queue holds one job on average, at \(\rho = 0.9\) it holds nine, at \(\rho = 0.99\) it holds ninety-nine.
This sourceStandard derivation of the M/M/1 queue, giving L = ρ/(1−ρ) and the 1/(1−ρ) blow-up of queue length and delay as utilization approaches one.
- The nonlinear-interaction role is precisely this term: at \(\rho = 0.5\) the queue holds one job on average, at \(\rho = 0.9\) it holds nine, at \(\rho = 0.99\) it holds ninety-nine.
Domain-specific¶
- Customs-Clearance Delay
- The queueing substructure matters: clearance facilities are constrained-capacity service stations, and arrival-rate spikes — post-holiday parcel surges, sudden tariff-schedule changes, emergency-response shipments — produce non-linear queue blow-up disproportionate to the shortfall in clearance throughput
This sourceThe standard reference for the equilibrium queueing results behind this convex delay growth; only the volume's contents, not the relevant passage, could be consulted for this check, and the customs-specific surge behaviour is not treated there.
- The queueing substructure matters: clearance facilities are constrained-capacity service stations, and arrival-rate spikes — post-holiday parcel surges, sudden tariff-schedule changes, emergency-response shipments — produce non-linear queue blow-up disproportionate to the shortfall in clearance throughput
- Offered load
- Random Variable
- Probability theory and statistical inference — the foundational object: estimators, test statistics, p-values, likelihood ratios, and posteriors are all random variables, and the X-versus-x discipline plus sampling distributions are the working apparatus. Finance and risk — returns, portfolio P&L, default indicators, and value-at-risk as a quantile are random variables, with copulas and stochastic processes built on top. Operations research and queueing — arrival times, service times, and queue lengths are modeled as random variables with specified distributions (Poisson arrivals, exponential service)
This sourceThe standard development of interarrival times, service times and queue lengths as random variables with specified distributions, with Poisson arrivals and exponential service as the canonical M/M/1 case.
- Probability theory and statistical inference — the foundational object: estimators, test statistics, p-values, likelihood ratios, and posteriors are all random variables, and the X-versus-x discipline plus sampling distributions are the working apparatus. Finance and risk — returns, portfolio P&L, default indicators, and value-at-risk as a quantile are random variables, with copulas and stochastic processes built on top. Operations research and queueing — arrival times, service times, and queue lengths are modeled as random variables with specified distributions (Poisson arrivals, exponential service)
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Links previously used in the corpus¶
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- https://search.worldcat.org/title/1095854 ×1
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- https://www.wiley.com/en-us/Queueing+Systems%2C+Volume+1%3A+Theory-p-9780471491101 ×1
- https://www.wiley.com/en-us/Queueing+Systems%2C+Volume+I-p-9780471491101 ×1
- https://www.wiley.com/en-us/Queueing+Systems,+Volume+I-p-9780471491101 ×1
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