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Kendall-Lee notation for queuing systems

Short answer

Kendall-Lee notation is (a/b/c):(d/e/f) where a: arrival distribution, b: service distribution, c: number of servers, d: queue discipline, e: maximum system capacity, f: calling population size. For example, (M/M/1):(FCFS/∞/∞) represents a single-server queue with Poisson arrivals, exponential service, and infinite population.

Write the Kendall-Lee Notation for Queuing Systems. Explain with example.

Kendall–Lee notation is a compact six-part code that describes the main features of a queuing system. Kendall gave the first three symbols; Lee added the last three.

(a / b / c):(d / e / f)(a\,/\,b\,/\,c) : (d\,/\,e\,/\,f)
SymbolMeaningTypical codes
aArrival (inter-arrival time) distributionM (Poisson/exponential), D (constant), EkE_k (Erlang-k), GI (general independent)
bService (holding) time distributionM, D, EkE_k, G
cNumber of parallel servers1, 2, ..., N
dQueue disciplineFCFS/FIFO, LCFS/LIFO, SIRO (random), PR (priority), GD (general)
eMaximum number allowed in the system (waiting + in service)finite K or ∞
fSize of the calling population (sources)finite or ∞

When the last three are FCFS/∞/∞ they are often left out, so M/M/1 means (M/M/1):(FCFS/∞/∞)(M/M/1):(FCFS/\infty/\infty).

Meaning of the codes

  • M (Markovian): Poisson arrivals with rate λ\lambda, i.e. exponential inter-arrival times; or exponential service with mean 1/μ1/\mu. Memoryless.
  • D: every inter-arrival or service time is the same.
  • EkE_k: sum of k exponential phases; less random than M.
  • G/GI: any distribution.

Examples

  1. (M/M/1):(FCFS/∞/∞) – single public call office with Poisson callers and exponential call durations; callers wait in line, infinite population.
    • Utilisation ρ=λ/μ\rho = \lambda/\mu, mean number in system L=ρ/(1−ρ)L = \rho/(1-\rho), mean time in queue Wq=ρ/(μ−λ)W_q = \rho/(\mu-\lambda).
    • If λ=10\lambda = 10 calls/h and mean call = 3 min (μ=20\mu = 20/h): ρ=0.5\rho = 0.5, L=1L = 1, Wq=0.5/10W_q = 0.5/10 h =3= 3 min.
  2. (M/M/N):(FCFS/∞/∞) – Erlang delay (Erlang C) model: N operators in a call centre with queuing.
  3. (M/M/N):(GD/N/∞) – Erlang loss (Erlang B) model: N trunks, no waiting space, blocked calls lost.
  4. (M/M/N):(GD/N/S) – Engset model: finite number S of sources, no waiting.
  5. (M/D/1):(FCFS/K/∞) – router output link with fixed-size packets and a buffer of K packets.

The notation lets an engineer pick the correct formula (Erlang B, Erlang C, Engset, M/M/1) by simply reading the system description.

✓ Answer checked by Claude Opus 5.5 (high effort)

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