--- title: "Short note: Queuing theory in delay system" subject: "Telecommunications" chapter: 6 key: "semester_8/telecommunications:short-note-queuing-delay" source: https://www.bctengineer.com/q/semester_8/telecommunications/short-note-queuing-delay site: BCT Engineering --- # Short note: Queuing theory in delay system Telecommunications · Chapter 6: Telephone Traffic - Asked in: 2065 Magh (old course), 10 marks - Video: [DSS 17EC654 module 3 Queuing System or Delay System](https://www.youtube.com/watch?v=-OXWEs32-qo) — Basavaraj Chougula - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_8/telecommunications/short-note-queuing-delay - Whole chapter: https://www.bctengineer.com/md/semester_8/telecommunications/chapter-6 ## Question Write a short note on queuing theory in delay system. ## Short answer Delay systems let calls wait instead of being lost. Queueing theory uses models like M/M/1 and M/M/n (Erlang C) to compute wait probability, mean queue length, and delay. For M/M/1, L = ρ/(1-ρ), W_q = ρ/(μ-λ); for M/M/n, C(n,A) gives probability of wait, with mean wait W_q = C·h/(n-A). Little's law: L = λW. ## Answer A **delay system** (queueing system or lost-calls-delayed system) is one in which a call that finds all servers busy is not lost but **waits in a queue** until a server becomes free. Queueing theory gives the probability of waiting, the mean waiting time and the queue length. Examples: calls waiting for a register (dial tone delay), messages in a store-and-forward data network, callers held at a call centre, jobs waiting for an SPC processor. #### Elements of a queueing system ```text arrivals queue (buffer) n servers --lambda--> [ | | | | ] ---> [S1] [S2] ---> departures [..] [Sn] ``` 1. **Arrival process:** usually Poisson with rate $\lambda$. 2. **Service time distribution:** usually exponential with mean $h = 1/\mu$. 3. **Number of servers** $n$. 4. **Queue capacity:** infinite or finite. 5. **Queue discipline:** FIFO, LIFO, random, priority. 6. **Number of sources:** infinite or finite. **Kendall's notation** A/B/n/K/N/D: arrival distribution / service distribution / servers / capacity / sources / discipline. Example: M/M/1, M/M/n, M/D/1 (M = Markov/exponential, D = deterministic, G = general). #### Main results **Little's law** (holds for any queue): $$ L = \lambda W, \qquad L_q = \lambda W_q $$ **M/M/1 queue** ($\rho = \lambda/\mu < 1$): $$ \begin{aligned} P(k) &= (1-\rho)\rho^k \\ L &= \frac{\rho}{1-\rho}, \quad L_q = \frac{\rho^2}{1-\rho} \\ W &= \frac{1}{\mu-\lambda}, \quad W_q = \frac{\rho}{\mu-\lambda} \end{aligned} $$ **M/M/n queue (Erlang C formula).** With $A = \lambda h$ erlang offered to $n$ servers ($A < n$), the probability that a call has to wait is $$ C(n,A) = \frac{\frac{A^n}{n!}\frac{n}{n-A}}{\sum_{k=0}^{n-1}\frac{A^k}{k!} + \frac{A^n}{n!}\frac{n}{n-A}} $$ Mean waiting time of all calls and of delayed calls: $$ W_q = \frac{C(n,A)\,h}{n-A}, \qquad W_{d} = \frac{h}{n-A} $$ Probability of waiting longer than $t$: $$ P(W > t) = C(n,A)\,e^{-(n-A)t/h} $$ #### Grade of service in delay systems The GoS is stated as the probability of delay, or the probability of waiting more than a given time, e.g. "dial tone within 3 s for 99% of calls". Delay systems give higher server utilisation than loss systems, but need $A < n$ for stability; as $\rho \to 1$ the waiting time grows very rapidly. #### Example An M/M/1 server with $\lambda = 8$/min and $\mu = 10$/min has $\rho = 0.8$, $L = 4$, $W = 1/(10-8) = 0.5$ min and $W_q = 0.4$ min. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In a delay system, a call that finds all servers busy: A. is lost and cleared B. waits in a queue until a server is free C. is routed to a final route automatically D. is accepted by an extra server Answer: B. Unlike a loss system, a delay system queues the call. (Key point: Delay system) 2. In M/M/1, the letters M, M and 1 stand for: A. Markov arrivals, Markov (exponential) service, one server B. Mean arrival, mean service, one queue C. Memory, memory, one buffer D. Multiple arrivals, multiple services, one customer Answer: A. M = Markov (Poisson arrivals/exponential service), and 1 is the number of servers. (Key point: Kendall notation) 3. Little's law L = lambda W holds for: A. only M/M/1 queues B. only exponential service times C. only finite queues D. any queue Answer: D. Little's law is general and does not depend on the arrival or service distribution. (Key point: Little's law) 4. For lambda = 8/min and mu = 10/min, the mean time in the system W is: A. 0.4 min B. 0.5 min C. 0.8 min D. 2 min Answer: B. W = 1/(mu - lambda) = 1/2 = 0.5 min; the queue wait Wq is 0.4 min. (Key point: M/M/1 example) ## Related in Telecommunications - [Queuing theory in delay systems](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Aqueuing-theory-delay-systems) — Telecommunications, chapter 6 - [Queuing system and Kendall's notation](https://www.bctengineer.com/question.md?key=semester_8%2Ftelecommunications%3Aqueuing-system-kendall) — Telecommunications, chapter 6, asked 4× ## Related in other subjects - [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 system usefulness and types](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Aqueuing-types-usefulness) — Simulation and Modeling, chapter 4 - [Short note: Application of queuing system](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Ashort-note-queuing-application) — Simulation and Modeling, chapter 4 - [Queuing block diagram; M/M/4 and D/M/2](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Aqueuing-block-diagram-mm4) — Simulation and Modeling, chapter 4