--- title: "Queuing notation, default values, elements" subject: "Simulation and Modeling" chapter: 4 key: "semester_8/simulation_and_modeling:queuing-notation-defaults-elements" source: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/queuing-notation-defaults-elements site: BCT Engineering --- # Queuing notation, default values, elements Simulation and Modeling · Chapter 4: Queuing system - Asked in: 2074 Magh, 5+5 marks - Video: [Kendall's Notation in Queueing Theory | Operation Research in hindi](https://www.youtube.com/watch?v=O3vfHQSAe0s) — Start Practicing - 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-notation-defaults-elements - Whole chapter: https://www.bctengineer.com/md/semester_8/simulation_and_modeling/chapter-4 ## Question Explain the queuing notation with at least two examples and also explain the possibility of default values for notation. Explain the different elements of a queue system. ## Short answer Kendall notation A/B/c/K/N/Z: A inter-arrival distribution, B service distribution, c servers, K capacity, N population, Z discipline (defaults: infinite capacity/population, FCFS). Example: M/M/1 (Poisson arrivals, exponential service, 1 server). Elements: calling population, arrival process, queue, service mechanism, discipline. ## Answer #### Queuing (Kendall) notation A queue is written as `A/B/c/K/N/D`: | Field | Meaning | |---|---| | A | Distribution of inter-arrival times | | B | Distribution of service times | | c | Number of parallel servers | | K | Maximum number of customers in the system (capacity) | | N | Size of the calling population | | D | Queue discipline | Codes for A and B: **M** = exponential (Markovian, Poisson arrivals), **D** = deterministic, **E_k** = Erlang-k, **G** = general. **Example 1: M/M/1.** Poisson arrivals, exponential service, one server. Measures with arrival rate `lambda` and service rate `mu`: utilization `rho = lambda/mu`, average number in system `L = rho/(1 - rho)`, average time in system `W = 1/(mu - lambda)`. For `lambda = 4`, `mu = 6` per hour: `rho = 0.667`, `L = 2`, `W = 0.5` hour. **Example 2: M/G/3/30/infinity/FCFS.** Poisson arrivals, general service-time distribution, 3 servers, at most 30 customers in the system, infinite population, first come first served. #### Default values The last three fields are often omitted. When omitted, the defaults are: - K (capacity) = infinity. - N (population) = infinity. - D (discipline) = FCFS. So `M/M/2` means `M/M/2/infinity/infinity/FCFS`, and `M/D/1/20` means capacity 20 with infinite population and FCFS. A field can be left out only from the right; the first three fields must always be given. #### Elements of a queue system ```text Population --> Arrival --> Queue --> Server(s) --> Departure ``` 1. **Calling population:** the group from which customers come. It may be infinite (public) or finite (machines in a factory). 2. **Arrival process:** the pattern of arrival, described by the inter-arrival time distribution and its mean rate, single or batch. 3. **Queue (waiting line):** where customers wait. It may have limited or unlimited capacity. 4. **Queue discipline:** the rule for selecting the next customer, e.g. FCFS, LIFO, priority or random. 5. **Service mechanism:** the server or servers, the service-time distribution and the arrangement (parallel or series). 6. **Departure:** customers leaving after service (or after balking or reneging). 7. **Customer behaviour:** balking, reneging and jockeying. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In A/B/c/K/N/D, what does N denote? A. Size of the calling population B. Number of servers C. Maximum customers in the system D. Queue discipline Answer: A. N is the calling population size; K is the capacity. (Key point: Notation fields) 2. What does M/M/2 mean when written in full? A. M/M/2/2/2/FCFS B. M/M/2/infinity/infinity/FCFS C. M/M/2/infinity/1/LIFO D. M/M/2/0/infinity/priority Answer: B. Omitted last three fields default to infinite capacity, infinite population and FCFS. (Key point: Default values) 3. Which fields may be omitted from Kendall notation? A. Any field in any position B. Only the first field A C. Only fields from the right end; the first three must always be given D. Only the middle fields K and N, but not D Answer: C. A field can be left out only from the right, so A, B and c are always required. (Key point: Omitting fields) 4. For M/M/1 with lambda = 4 and mu = 6 per hour, what are rho and L? A. rho = 1.5, L = 3 B. rho = 0.667, L = 0.5 C. rho = 0.333, L = 2 D. rho = 0.667, L = 2 Answer: D. rho = 4/6 = 0.667 and L = rho/(1 - rho) = 0.667/0.333 = 2. 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