Queuing theory
- 1. Queuing Theory Y. B. Kumbhar Ybk_aero@adcet.in
- 4. Queue Parameters • Calling Population • Arrival Process • Queue Discipline • Service Process • Number of Servers
- 5. 5 1. The Calling Population • Population of customers or jobs • The size can be finite or infinite – The latter is most common • Can be homogeneous – Only one type of customers/ jobs • Or heterogeneous – Several different kinds of customers/jobs
- 6. 6 2. Arrival Process • In what pattern do jobs / customers arrive to the queueing system? – Distribution of arrival times? – Batch arrivals? – Finite population? – Finite queue length? • Poisson arrival process often assumed – Many real-world arrival processes can be modeled using a Poisson process
- 7. 7 3. Service Process • How long does it take to service a job or customer? – Distribution of arrival times? – Rework or repair? – Service center (machine) breakdown? • Exponential service times often assumed – Works well for maintenance or unscheduled service situations
- 8. 8 4. Number of Servers • How many servers are available? Single Server Queue Multiple Server Queue
- 9. 9 Example – Two Queue Configurations Servers Multiple Queues Servers Single Queue
- 10. 10 1.The service provided can be differentiated – Ex. Supermarket express lanes 2. Labor specialization possible 3. Customer has more flexibility 4. Balking behavior may be deterred – Several medium-length lines are less intimidating than one very long line 1. Guarantees fairness – FIFO applied to all arrivals 2. No customer anxiety regarding choice of queue 3. Avoids “cutting in” problems 4. The most efficient set up for minimizing time in the queue 5. Jockeying (line switching) is avoided Multiple vs Single Customer Queue Configuration Multiple Line Advantages Single Line Advantages
- 11. 11 5. Queue Discipline • How are jobs / customers selected from the queue for service? – First Come First Served (FCFS) – Shortest Processing Time (SPT) – Earliest Due Date (EDD) – Priority (jobs are in different priority classes) • FCFS default assumption for most models
- 12. Single-server Single-stage Queue Service Facility Customers In queue Arrival Stream
- 14. Single-server Multiple-stage Queue Service Facility Customers In queue Pharmacy Conveyor System >>>>>
- 16. Queuing Models (P/Q /R):(X/Y/Z) P : arrival rate distribution Q : Service rate distribution R : Number of servers X : Service Discipline Y : Max. customer permitted in system Z : Size of calling population
- 17. Model 1: (M/M/1):(GD/∞/∞) • M: Poisson arrival rate • M: Poisson service rate • 1: 1 Service server • GD : General Discipline • ∞ : Infinite customers are allowed in the system • ∞ : Customers are called from an infinite polulation
- 18. Other models • Model 2: ( M / M / C) : ( G D / ∞ / ∞ ) - C number of servers • Model 3: ( M / M / 1 ) : ( G D / N/ ∞ ) - N customer allowed in system • Model 4: ( M / M / C) : ( G D / N/ ∞ ) • Model 5: ( M / M / 1 ) : ( G D / N/ N) - calling population is finite (N) • Model 6: ( M / M / C) : ( G D / N/ N)
- 19. Formulae for Model 1(M/M/1):(GD/∞/∞) • λ=arrival rate per unit time • μ=service rate per unit time • Utilization factor Φ= λ μ • Probability that there are n customers in the system, • 𝑃 𝑛 = (1 - φ) φ 𝑛 n=1,2,3…
- 20. Formulae for Model 1 (M/M/1):(GD/∞/∞)—Contd. • Number of customers in a system, 𝐿 𝑠 = φ (1 − φ) • Number of customers in a queue, 𝐿 𝑞 = φ2 (1 − φ)
- 21. Formulae for Model 1 (M/M/1):(GD/∞/∞)—Contd. • Average Waiting time of customers in a system, 𝑊𝑠 = 1 (μ − λ) • Average Waiting time of customers in a queue, 𝑊𝑞 = φ (μ − λ)