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Kendall's notation

Kendall's notation is a compact symbolic system used in queueing theory to classify and describe the essential characteristics of queueing models, such as arrival processes, service mechanisms, and system capacities. Introduced by British mathematician David G. Kendall in his 1953 paper on stochastic processes in queues, the notation provides a standardized shorthand that facilitates analysis and comparison of diverse queueing systems across operations research and applied mathematics.^[1] In its basic form, Kendall's notation is expressed as A/B/c, where A represents the probability distribution of interarrival times (e.g., M for Markovian or Poisson arrivals with exponential interarrivals, D for deterministic constant intervals, or G for general distributions), B denotes the service time distribution (similarly, M for exponential, D for deterministic, or G for general), and c indicates the number of parallel servers (an integer, often 1 for single-server models).^[2] This trio captures the core stochastic behavior of a queue, enabling precise modeling of phenomena like waiting lines in telecommunications, manufacturing, and service industries.^[3] The notation has been extended over time to A/B/c/K/N/D for more complex systems, where K specifies the system's capacity (maximum queue length plus servers, defaulting to infinity if omitted), N denotes the finite population size from which arrivals draw (defaulting to infinity), and D describes the queue discipline (e.g., FCFS for first-come, first-served, defaulting to FCFS if unspecified).^[4] These extensions, building on Kendall's foundational work, accommodate variations like bounded queues or limited caller populations, enhancing applicability to real-world scenarios such as emergency departments or computer networks. Common examples include the M/M/1 queue for a single-server system with Poisson arrivals and exponential service times, and M/M/c for multi-server variants, both of which assume infinite capacity and population.^[5]

Introduction

Definition and Purpose

Queueing theory is a branch of operations research and applied probability that models systems where customers or jobs arrive randomly, may need to wait due to limited service capacity, and then receive service before departing. These systems are prevalent in everyday scenarios, such as bank teller lines or computer networks, where the goal is to analyze waiting times, queue lengths, and resource utilization to optimize performance.^[6] Kendall's notation provides a standardized symbolic framework for describing queueing systems, typically represented as A/B/c/K/N/D, where each component denotes a particular characteristic of the model. Introduced by mathematician David G. Kendall in 1953, this notation captures essential elements like the nature of arrivals, service mechanisms, and system constraints in a compact form.^[1]^[7] The primary purpose of Kendall's notation is to enable precise and concise specification of stochastic processes in queueing models, allowing researchers and practitioners to communicate complex system behaviors without extensive prose. This standardization simplifies the comparison of different queue configurations, supports mathematical analysis for deriving performance metrics like average wait times, and aids in simulation studies for system design.^[7]^[3] Since its establishment in 1953, Kendall's notation has become a foundational tool in operations research, computer science, and telecommunications, where it facilitates modeling of diverse applications from manufacturing lines to network traffic management. Its widespread adoption promotes consistency in literature and practice, enhancing collaboration across disciplines.^[7]

Historical Development

Kendall's notation for describing queueing systems was first proposed by British mathematician David G. Kendall in his seminal 1953 paper, where he introduced the compact form A/B/c to classify models based on arrival process (A), service time distribution (B), and number of servers (c).^[1] This notation emerged as a standardized shorthand amid growing interest in stochastic processes for analyzing waiting lines, providing a clear way to denote system characteristics without lengthy descriptions.^[1] The development of Kendall's notation occurred during the post-World War II surge in operations research and stochastic modeling, a period marked by expanded applications of probability theory to real-world problems like telecommunications and logistics.^[5] It built directly on the foundational work of Danish engineer Agner Krarup Erlang, whose early 20th-century models for telephone traffic engineering laid the groundwork for queueing analysis by introducing concepts like the Erlang distribution for call arrivals and holding times.^[5] Kendall's contribution advanced this legacy by formalizing a general framework for more complex, non-deterministic systems. Over the following decade, the notation evolved from its initial A/B/c structure to incorporate additional parameters for system capacity (K), population size (N), and queue discipline (D), forming the extended A/B/c/K/N/D form by the mid-1960s.^[8] This expansion, often referred to as the Kendall-Lee notation, was facilitated by works like Alec M. Lee's 1966 text on applied queueing theory, which integrated these elements to handle finite resources and varied service rules.^[8] Kendall himself further influenced this progression through his research on birth-death processes, which modeled queue length changes as continuous-time Markov chains and became central to analyzing multi-server systems.^[1] By the 1970s, Kendall's notation had become the de facto standard in queueing literature, as evidenced by its prominent use in Leonard Kleinrock's influential 1975 textbook Queueing Systems Volume I: Theory, which refined and popularized the framework for computer and communication networks. As of 2025, the notation remains a cornerstone of queueing theory, with only minor extensions in areas like simulation software for handling hybrid or adaptive disciplines, ensuring its enduring utility without fundamental changes.^[9]

Core Parameters

Arrival Process (A)

In Kendall's notation, the first parameter, A, describes the stochastic process governing customer arrivals, typically specifying the distribution of inter-arrival times between consecutive customers entering the queueing system.^[1] Common symbols for A include M for Markovian (or memoryless) arrivals, which model a Poisson process with exponentially distributed inter-arrival times at rate λ; D for deterministic arrivals at fixed intervals; G for a general arbitrary distribution; Ek for the Erlang-k distribution, representing the sum of k independent exponential inter-arrival times; and PH for phase-type distributions, which approximate more complex distributions through a finite number of exponential phases.^[10]^[11] For the M case, the number of arrivals in a time interval t follows a Poisson distribution, with probability mass function

P(N(t) = n) = \frac{(\lambda t)^n e^{-\lambda t}}{n!}, \quad n = 0, 1, 2, \dots

where λ is the mean arrival rate.^[10] The arrival process is generally assumed to consist of independent and identically distributed inter-arrival times, forming a stationary renewal process, with an infinite population of potential customers by default unless the population size N is explicitly finite in the notation.^[10] This parameter directly influences the system's traffic intensity, defined as ρ = λ / (c μ), where c is the number of servers and μ is the service rate per server, providing a key measure of utilization and stability when combined with the service time distribution.^[10]

Service Time Distribution (S)

In Kendall's notation, the second parameter, denoted as S, specifies the probability distribution governing the time required to serve each customer at a server. This distribution characterizes the stochastic nature of service completion times in the queueing system.^[12] Service times are typically assumed to be independent of arrival times and identically distributed across all customers and servers, unless the model specifies otherwise. Common notations for S include M for the exponential (Markovian) distribution with mean service rate \mu, where the probability density function is given by
f(t) = \mu e^{-\mu t}, \quad t \geq 0;
D for deterministic service times that are constant and non-random; G for a general arbitrary distribution without further specification; H for the hyperexponential distribution, which models highly variable service times as a mixture of exponentials; and E_k for the Erlang-k distribution, a gamma distribution with shape parameter k representing k phased exponential services.^[13]^[2] The choice of S significantly influences key performance measures, such as the server utilization \rho = \lambda / (c \mu) in systems with exponential service (where \lambda is the arrival rate and c is the number of servers), and it impacts waiting times—for instance, in single-server queues with general service, via the Pollaczek-Khinchine formula relating mean queue length to the service time variance.^[13]^[10]

Number of Servers (c)

In Kendall's notation, the parameter c specifies the finite number of identical servers operating in parallel to handle arriving customers, forming the third component of the standard A/S/c descriptor for a queueing system. This notation, proposed by David G. Kendall, allows for the concise classification of queue configurations, with c quantifying the parallel processing capacity.^[1] By convention, c = 1 describes single-server queues, where all customers are processed sequentially by one server, while c > 1 models multi-server systems enabling simultaneous service of multiple customers from a shared queue. The servers are assumed to be homogeneous, each governed by the identical service time distribution S from the notation's second parameter, with the standard model excluding server breakdowns or vacations unless explicitly extended.^[1]^[14] Multiple servers elevate the system's aggregate service rate to c \mu, where \mu is the individual server's mean service rate determined by S, thereby enhancing throughput compared to single-server setups. For infinite queues, steady-state stability requires the traffic intensity \rho = \frac{\lambda}{c \mu} < 1, with \lambda as the arrival rate; violation of this condition leads to unbounded queue growth.^[14] Analysis of multi-server queues (c > 1) differs markedly from the single-server M/M/1 case, as performance measures like the probability of delay incorporate the Erlang-C formula to capture the dynamics of queueing when all servers are occupied, rather than relying on basic birth-death process solutions. This formula, rooted in early telephony models, provides the proportion of arrivals experiencing wait time and underpins metrics such as average queue length in M/M/c systems.^[14]

Capacity and Population Parameters

System Capacity (K)

In Kendall's notation for queueing systems, the parameter K denotes the total capacity of the system, defined as the maximum number of customers that can simultaneously be present, including those waiting in the queue and those receiving service. This capacity is the sum of the queue length and the number of occupied servers, with [K](/page/K) required to be at least as large as the number of servers c (i.e., K \geq c). By convention, if no explicit limit is specified, K = \infty, permitting an unbounded queue.^[9] A finite K introduces blocking, where customer arrivals are rejected if the system is full, resulting in lost customers and a positive blocking probability equal to the steady-state probability \pi_K that the system is at capacity. This contrasts with infinite-capacity models by preventing queue growth without bound and altering performance metrics, such as throughput, which becomes the effective arrival rate \lambda (1 - \pi_K). When K = c, the system has no queue space and operates as a pure loss system, exemplified by the M/M/c/c model, where all blocked arrivals are immediately lost without waiting.^[15] For single-server Markovian systems analyzed via birth-death processes, the steady-state distribution with finite K truncates the infinite-state solution. In the single-server case (c=1), letting \rho = \lambda / \mu, the probabilities are \pi_n = \pi_0 \rho^n for n = 0, 1, \dots, K, where \pi_0 = (1 - \rho) / (1 - \rho^{K+1}) for \rho \neq 1, and the blocking probability is \pi_K = \rho^K (1 - \rho) / (1 - \rho^{K+1}), or equivalently \pi_K = \rho^K / \sum_{n=0}^K \rho^n. For \rho = 1, \pi_n = 1/(K+1) uniformly. These expressions derive from balancing the birth and death rates in the truncated state space. For multi-server systems (c > 1), the steady-state probabilities follow a similar truncation but use state-dependent service rates, resulting in a more complex piecewise distribution.^[3]^[15] A prominent example is the M/M/1/K queue, a single-server system with finite buffer capacity K-1, where arrivals follow a Poisson process at rate \lambda and service times are exponential at rate \mu. This model captures scenarios with limited resources, such as disk storage or memory buffers, where exceeding capacity leads to overflow. The finite K yields higher blocking probabilities than the infinite M/M/1 case, especially under heavy load, making it suitable for performance evaluation in constrained environments like computer networks with finite packet buffers.^[16]

Calling Population (N)

In Kendall's notation for queueing systems, the parameter N represents the size of the calling population, which is the total number of potential customers available to arrive at the system. By default, N = \infty, corresponding to an open queueing system where the arrival process is independent of the system's current state, as the pool of customers is effectively unlimited.^[17] When N is finite, the model describes a closed queueing system, in which the arrival rate becomes state-dependent because the number of potential customers decreases as more enter the system. A classic example is the machine repair model, where N machines each generate repair requests independently, but the overall arrival rate to the repair facility diminishes as more machines are already queued or under repair. In such cases, the arrival rate \lambda(n) when there are n customers in the system is given by \lambda(n) = (N - n) \gamma, where \gamma is the individual arrival rate of each potential customer outside the system. This dependence bounds the arrivals, preventing the system from experiencing the unbounded growth possible in open models.^[17]^[18] Common notations for finite N include M/M/c/N/N, which specifies a finite population of size N with system capacity limited to N (implying no additional arrivals once the system is full, and often no explicit queue beyond servers if capacity aligns with population). For multi-queue scenarios, finite N appears in closed queueing networks analyzed via the Gordon-Newell theorem, where customers circulate among nodes without external arrivals, enabling product-form steady-state distributions under exponential service assumptions.^[17] Performance analysis for finite N typically employs mean value analysis (MVA), an iterative method that computes key metrics like average queue lengths and response times by incrementally adding customers to the network, leveraging the state-dependent rates to derive throughput and utilization without solving full state probabilities. Unlike open queues with constant arrival rates, finite N ensures saturation effects, where performance plateaus as N increases, providing bounded measures such as expected system occupancy L = \sum_{n=0}^{N} n P_n, with P_n derived from birth-death processes adjusted for \lambda(n). This contrast highlights how finite populations model resource-constrained environments, like computer systems or manufacturing lines, more realistically than infinite assumptions.^[17]

Queue Discipline and Extensions

Queue Discipline (D)

In Kendall's notation for queueing systems, the parameter D denotes the queue discipline, which specifies the policy for selecting the next customer from the queue to receive service. This parameter defines the ordering mechanism, ensuring that servers remain busy whenever customers are present in a work-conserving manner.^[19]^[20] The default and most common discipline is FCFS (first-come, first-served), where customers are served in the order of their arrival; this is often omitted from the notation when it applies. Other standard disciplines include LCFS (last-come, first-served), in which the most recent arrival is served next; SIRO (service in random order), where selection is random among waiting customers; and priority-based systems, which can be preemptive (interrupting ongoing service) or non-preemptive (completing current service before switching), typically denoted by P in extended notation. Shortest job first (SJF) is another notable discipline, prioritizing customers with the shortest expected service time.^[19]^[20]^[21] Queue disciplines are generally assumed to be non-preemptive unless explicitly stated otherwise, meaning a customer's service is not interrupted once started, and no jockeying occurs between servers (i.e., customers do not switch lines mid-process). For work-conserving disciplines that do not rely on knowledge of individual service times—such as FCFS, LCFS, and SIRO—the mean waiting time remains independent of the specific policy, as governed by conservation laws in queueing theory; however, the distribution of waiting times varies significantly. Little's law, relating the average number of customers to the arrival rate and mean time in the system, continues to hold across these policies. In particular, FCFS minimizes the variance of waiting times compared to alternatives like LCFS or SIRO, promoting fairness in stochastic environments.^[20]^[22] Disciplines like SJF or priority queues, which incorporate service time estimates or customer classes, can alter both the mean and variance of waiting times; for instance, SJF reduces overall mean waiting time and exhibits lower variance than FCFS by favoring shorter jobs, though it may increase waits for longer ones. These effects highlight how D influences system performance beyond basic throughput, guiding applications where fairness, efficiency, or prioritization is critical.^[22]^[23]

Variations and Extensions

While Kendall's notation effectively characterizes basic single-queue systems with identical servers and first-in-first-out discipline, it assumes a single waiting line and does not inherently account for heterogeneous servers, multiple queues, or advanced service rules like priorities, limiting its direct application to complex real-world scenarios such as telecommunication networks or manufacturing lines. These limitations become evident in systems where customer classes require differentiated treatment or where queues interact, necessitating extensions to the standard A/S/c/K/N/D framework.^[24] To address priorities, the queue discipline parameter D is often extended to specify "PR" for priority service, where higher-priority customers preempt or overtake lower-priority ones within the same queue, as formalized in early extensions for multi-class systems.^[24] For batch arrivals or services, superscripts are appended to the arrival (A) or service (S) descriptors, such as M^X/M/1 to denote Poisson batch arrivals with random group size X following a general distribution, commonly applied in transportation or production settings where entities arrive or are processed in groups. Additionally, phase-type (PH) distributions replace M or G in A or S positions to model more flexible, non-Markovian behaviors as sums of exponential phases, enabling tractable analysis of hyperexponential or Erlang-like processes in computational queueing models. Post-2000 developments in matrix-analytic methods have introduced notations inspired by Kendall's framework for solving intricate queues, particularly quasi-birth-death (QBD) processes, which generalize birth-death chains to multilevel structures with phase-type transitions and are denoted in extended forms like PH/PH/1 analyzed via matrix-geometric solutions for stability and performance. These QBD models, pivotal since Neuts' foundational work, facilitate the study of virtual queues in cloud computing environments, where resource allocation mimics multi-server queues with phase-dependent arrivals to optimize latency in data centers.^[25] For queueing networks, Kendall's notation applies to individual nodes (e.g., M/M/1 per station), but the overall system lacks a unified descriptor; Burke's theorem establishes that in tandem M/M/c queues, the departure process from each station is Poisson with rate equal to the arrival rate, enabling independent analysis of series configurations like assembly lines. Similarly, Jackson networks extend this to open networks of M/M/c nodes with product-form stationary distributions, where joint queue lengths factorize as if queues were independent, a high-impact result for modeling routing in communication systems. Criticisms highlight that standard Kendall's notation inadequately captures retrial mechanisms, where blocked customers join an "orbit" and retry independently, as it omits the orbit's time distribution; contemporary texts advocate additional symbols beyond D to denote retrials (e.g., specifying orbit dynamics separately), addressing gaps in applications like call centers with repeated attempts. As of 2025, these updates in queueing network analyses, such as those incorporating retrials and priorities, enhance applicability to dynamic systems like cloud orchestration, though no universal extension has fully standardized them.

Examples and Applications

Single-Server Infinite Queue (M/M/1)

The M/M/1 queue exemplifies the simplest form of Kendall's notation for a queueing system, where the arrival process (A) is Markovian (M), indicating Poisson arrivals at rate λ; the service time distribution (S) is also Markovian (M), meaning exponentially distributed service times with rate μ; the number of servers (c) is 1; the system capacity (K) is infinite; the calling population (N) is infinite; and the queue discipline (D) is typically first-come, first-served (FCFS).^[1] This configuration models scenarios with random arrivals and service completions, memoryless due to the exponential distributions.^[26] For stability, the system requires the traffic intensity ρ = λ/μ < 1, ensuring the arrival rate does not overwhelm the service capacity, leading to a steady-state distribution.^[1] The M/M/1 queue is analyzed as a continuous-time birth-death process, with birth rate λ_n = λ for all n ≥ 0 and death rate μ_n = μ for n ≥ 1 (μ_0 = 0).^[27] The steady-state probabilities π_n, representing the long-run proportion of time the system has n customers, satisfy the balance equations λ π_{n-1} = μ π_n for n ≥ 1, with ∑ π_n = 1. Solving yields π_n = (1 - ρ) ρ^n for n = 0, 1, 2, ..., where π_0 = 1 - ρ is the probability of an idle server.^[26] Key performance metrics derive from these probabilities. The average number of customers in the queue (excluding the one in service) is L_q = ∑_{n=1}^∞ (n-1) π_n = ρ^2 / (1 - ρ).^[18] The average waiting time in the queue is W_q = L_q / λ = ρ / (μ (1 - ρ)), obtained via Little's law relating queue length to arrival rate and waiting time.^[27] These formulas highlight how utilization ρ drives congestion: as ρ approaches 1, L_q and W_q grow unbounded, emphasizing the need for ρ < 1.^[26] The M/M/1 model applies to basic single-agent operations, such as a simple call center with one operator handling incoming calls that arrive randomly and require exponentially distributed handling times.^[28] For instance, in a support hotline with λ = 2 calls per minute and μ = 3 services per minute (ρ = 2/3), the expected queue length is L_q = (2/3)^2 / (1 - 2/3) = 4/3 customers, illustrating manageable wait times under stable conditions.^[18]

Multi-Server Queue (M/M/c)

The M/M/c queue denotes a queueing system in Kendall's notation where arrivals follow a Poisson process with rate λ (A=M), service times are exponentially distributed with rate μ per server (S=M), there are c finite servers with c > 1 (c), the system capacity is infinite (K=∞), the calling population is infinite (N=∞), and customers are served in first-come, first-served order (D=FCFS). This model captures scenarios where multiple parallel servers handle demands, such as call centers or checkout lanes, assuming no balking or reneging. The foundational notation was introduced by Kendall to standardize descriptions of such stochastic processes.^[12] For stability, the traffic intensity must satisfy ρ = λ / (c μ) < 1, ensuring the long-run arrival rate does not exceed the maximum service capacity; otherwise, the queue length grows unbounded. A key performance metric is the probability that all c servers are busy upon arrival (Erlang C formula), which determines the likelihood of queuing delay:

C(c, a) = \frac{\frac{a^c}{c!}}{\sum_{k=0}^{c-1} \frac{a^k}{k!} + \frac{a^c}{c!} \cdot \frac{1}{1 - \rho}}

where a = λ / μ is the offered load. This formula arises from the steady-state probabilities of a birth-death process state diagram, where birth rates are constant at λ and death rates are min(n, c) μ for n customers in the system, though full derivations involve normalizing the infinite series for queue states. The average number of customers in the queue is then

L_q = C(c, a) \frac{\rho}{1 - \rho}.

This expression highlights how queuing builds only when all servers are occupied, with the factor ρ / (1 - ρ) reflecting the geometric tail of excess arrivals.^[29] Compared to the single-server M/M/1 case, the M/M/c model reduces average waiting times through server parallelism, as arriving customers can be routed to any idle server, mitigating bottlenecks from simultaneous demands. It finds practical application in supermarkets, where multiple cashiers serve Poisson-like customer flows to minimize checkout delays and optimize staffing. The Erlang C framework originated from A. K. Erlang's early 20th-century telephony analyses but was adapted for general multi-server queues in modern queueing theory.^[29]

Finite Capacity Queue (M/M/1/K)

The finite capacity queue in Kendall's notation is represented as M/M/1/K, where arrivals follow a Poisson process (A=M), service times are exponentially distributed (S=M), there is one server (c=1), the system capacity is finite at K (including the server), the calling population is infinite (N=∞), and the queue discipline is first-come, first-served (D=FCFS).^[1]^[9] This model extends the basic single-server queue by imposing a strict limit on the total number of customers that can be accommodated, leading to blocking of new arrivals when the system reaches capacity K.^[30] In the M/M/1/K queue, the system behaves as a birth-death process where the arrival rate λ is zero when the number of customers n = K, preventing further entries and causing potential losses.^[9] The effective throughput of the system is thus reduced to λ (1 - π_K), where π_K is the probability that the system is full, reflecting the fraction of arrivals that successfully enter.^[30] Unlike the infinite-capacity M/M/1 queue, which assumes unbounded waiting room, the finite K introduces saturation effects that bound queue lengths but increase the risk of customer rejection under high load (ρ = λ/μ > 1).^[9] The steady-state probabilities for the number of customers n (0 ≤ n ≤ K) are given by

\pi_n = \frac{(1 - \rho) \rho^n}{1 - \rho^{K+1}}, \quad \rho \neq 1,

with the special case π_n = 1/(K+1) when ρ = 1; these form a truncated geometric distribution normalized over the finite state space.^[9]^[30] The blocking probability, which quantifies the proportion of arrivals turned away, is π_K = \frac{(1 - \rho) \rho^K}{1 - \rho^{K+1}} for ρ ≠ 1, or 1/(K+1) otherwise.^[9] Key performance metrics include the average number of customers in the system L = \sum_{n=0}^K n \pi_n, which can be computed explicitly as L = \frac{\rho (1 - (K+1) \rho^K + K \rho^{K+1}) }{(1 - \rho) (1 - \rho^{K+1})} for ρ ≠ 1.^[30] The average queue length (excluding the server) is L_q = L - (1 - \pi_0), where \pi_0 is the steady-state probability that the system is empty. This accounts for the server utilization being 1 - \pi_0.^[31] These measures highlight how finite capacity trades off reduced waiting times for higher blocking under overload, with stability guaranteed regardless of ρ due to the bounded states.^[30] The M/M/1/K model finds applications in buffer-limited networks, such as telecommunication systems where finite packet buffers prevent excessive delays but lead to packet drops during congestion.^[32]^[33] In these contexts, it aids in optimizing buffer sizes to balance throughput and loss rates, as seen in analyses of network flow dynamics and queueing in overloaded communication links.^[34]

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[29]
[PDF] The Erlang B and C Formulas
The hrlang B and C formulas are true probability classics. Indeed, much of the theory was developed by A. K. hrlang and his colleagues prior to 1925; ...
[30]
M/M/1/K Queueing Model | Real Statistics Using Excel
Describes the M/M/1/K queueing model and provides formulas for various characteristics of this model, and explains how to calculate them in Excel.
[31]
[PDF] A Queueing Approach to Model Network Flow Dynamics
Aug 31, 2006 · 42. Figure 18 confirms that the M/M/1/K queue is a suitable model to clarify network rejections. The good match again ...
[32]
[PDF] Analysis of Packet Queueing in Telecommunication Networks | UPF
This will be done in more detail for the M/M/1/K system. Performance Metrics. Refer to Section 5.8. 5.9.2 M/M/1/K queue.<|separator|>
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Buffer and throughput trade-offs in M/G/1/K queueing networks
More buffers will lead to a throughput that is generally higher than before; the monotonicity and concavity of the throughput with respect to buffer size (for ...