--- title: "Queuing model; M/D/2/60/150/FIFO meaning" subject: "Simulation and Modeling" chapter: 4 key: "semester_8/simulation_and_modeling:queuing-model-mdpq-kendall" source: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/queuing-model-mdpq-kendall site: BCT Engineering --- # Queuing model; M/D/2/60/150/FIFO meaning Simulation and Modeling · Chapter 4: Queuing system - Asked in: 2073 Magh, 3+2+5 marks - Video: [Queuing Theory Tutorial - Queues/Lines, Characteristics, Kendall Notation, M/M/1 Queues](https://www.youtube.com/watch?v=SqSUJ0UYWMQ) — Bikey Bonn Kleiford Seranilla - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/queuing-model-mdpq-kendall - Whole chapter: https://www.bctengineer.com/md/semester_8/simulation_and_modeling/chapter-4 ## Question What is a queuing model? Explain with figure. What is the meaning of M/D/2/60/150/FIFO in queuing notation? Explain the Kendall notation with example. ## Short answer A queuing model mathematically describes customers arriving, waiting, and being served, for predicting wait time and length. Figure: calling population -> queue -> server(s) -> departures. M/D/2/60/150/FIFO: Poisson arrivals, deterministic service, 2 servers, capacity 60, finite population 150, first-in first-out. Kendall notation A/B/c/K/N/Z: A inter-arrival dist, B service dist, c servers, K capacity, N population, Z discipline. Example: M/M/1 (exponential inter-arrival/service, 1 server). ## Answer #### Queuing model A **queuing model** is a mathematical and logical description of a system in which customers (jobs, packets, patients) arrive, wait in a line if the server is busy, get service and leave. It is used to predict measures such as average waiting time, queue length and server utilisation. Main elements of the model: - **Calling population**: the source of customers (finite or infinite). - **Arrival process**: how customers arrive, described by the inter-arrival time distribution (for example Poisson arrivals, mean rate lambda). - **Queue**: the waiting area, with a capacity limit and a discipline (FIFO, LIFO, priority). - **Service mechanism**: one or more servers with a service time distribution (mean rate mu). - **Departure**: customers leave after service. #### Figure ```text Arrivals +-----------------+ +--------+ Departures (calling --> | Queue (waiting | --> | Server | --> population) | line, buffer) | | (s) | +-----------------+ +--------+ queue discipline service (FIFO, LIFO...) time dist. ``` In a simulation, the clock advances from event to event (arrival, start of service, departure). The state is the number in the system and the server status. #### Meaning of M/D/2/60/150/FIFO The fields are read in the order Arrival / Service / Servers / System capacity / Population / Discipline. | Field | Value | Meaning | |---|---|---| | M | Markovian | Poisson arrivals, i.e. exponential inter-arrival times | | D | Deterministic | Constant (fixed) service time | | 2 | Servers | Two parallel servers | | 60 | System capacity | At most 60 customers in the system (queue plus those in service); others are turned away | | 150 | Calling population | Finite source of 150 customers | | FIFO | Discipline | First in, first out | Example: 150 registered users send print jobs. Jobs arrive randomly, two printers each take a fixed time per job, at most 60 jobs can be held, and jobs are printed in order of arrival. #### Kendall notation Kendall's notation describes a queue in a short code **A/B/c/K/N/Z** (Kendall originally used A/B/c; the other three fields were added later). If the last three fields are omitted, they are taken as infinite capacity, infinite population and FIFO. | Symbol | Position | Meaning | Common codes | |---|---|---|---| | A | 1st | Inter-arrival time distribution | M (exponential), D (deterministic), Ek (Erlang), G (general) | | B | 2nd | Service time distribution | M, D, Ek, G | | c | 3rd | Number of parallel servers | 1, 2, 3, ... | | K | 4th | System capacity (queue + service) | finite number or infinity | | N | 5th | Size of calling population | finite number or infinity | | Z | 6th | Queue discipline | FIFO, LIFO, SIRO, PRI | Example: **M/M/1** means Poisson arrivals, exponential service, one server, infinite capacity and population, FIFO. A single-teller bank counter is a typical M/M/1 queue. **M/M/3/20** is a three-server queue where at most 20 customers can be in the system. **M/G/1** has general service times, as in a CPU with arbitrary job lengths. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is a queuing model used to predict? A. Waiting time, queue length and server utilisation B. The hardware cost of servers C. The exact arrival time of each customer D. Only the number of servers needed with no other output Answer: A. A queuing model predicts performance measures like average wait, queue length and utilisation. (Key point: Queuing model) 2. In M/D/2/60/150/FIFO, what do D and 60 mean? A. D = exponential service; 60 = servers B. D = constant service time; 60 = maximum customers in the system C. D = deterministic arrivals; 60 = population D. D = discipline; 60 = mean service time Answer: B. The second field D is deterministic service and the fourth field 60 is the system capacity. (Key point: M/D/2/60/150/FIFO) 3. In M/D/2/60/150/FIFO, what is 150? A. 150 parallel servers B. The queue capacity excluding those in service C. A finite calling population of 150 customers D. The mean arrival rate per hour Answer: C. The fifth field is the calling population size. (Key point: Calling population) 4. What was Kendall's original notation and the defaults for omitted fields? A. Original A/B/c/K/N/Z; omitted fields mean zero capacity B. Original A/B only; omitted fields mean LIFO C. Original A/B/c; omitted fields mean finite capacity and SIRO D. Original A/B/c; omitted fields mean infinite capacity, infinite population and FIFO Answer: D. Kendall used A/B/c and the other three fields were added later, with infinite capacity, infinite population and FIFO as defaults. (Key point: Origin and defaults) ## Related in Simulation and Modeling - [Queuing model, Kendall, M/D/8/15/LIFO](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Aqueuing-model-mdlifo-8) — Simulation and Modeling, chapter 4 - [Queuing system, characteristics, Kendall notation](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Aqueuing-characteristics-role-kendall) — Simulation and Modeling, chapter 4, asked 2× - [Queuing characteristics; M/D/6/10/FIFO](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Aqueuing-characteristics-mdfifo-6) — Simulation and Modeling, chapter 4 - [Queuing notation, default values, elements](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Aqueuing-notation-defaults-elements) — Simulation and Modeling, chapter 4 ## Related in other subjects - [Queuing system and Kendall's notation](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Aqueuing-system-kendall) — Telecommunications, chapter 6, asked 4× - [Kendall-Lee notation for queuing systems](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Akendall-lee-notation) — Telecommunications, chapter 6, asked 2× - [150 km line by T-model, 60 MW](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3A150km-t-model-60mw) — Power System Analysis I, chapter 5 - [Erlang delay model: notation and derivation](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Aerlang-delay-kendall-derivation) — Telecommunications, chapter 6