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3-1-6 Queueing Theory
p. 53
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Queueing theory2p. 53 · PDF 57
3-1-6 Queueing Theory
| Term | Representation | Meaning (using customers at a store checkout counter as an example) |
|---|---|---|
| Average arrival rate | λ | The number of customers who arrive per unit time. |
| Average inter-arrival time | 1/λ | The average time interval (Ta) at which customers arrive at the checkout counter. |
| Average service rate | μ | The number of customers who can be processed per unit time. |
| Average service time | 1/μ | The average processing time (Ts) per customer. |
| Usage rate (utilization factor) | ρ | The checkout counter usage rate (the ratio of processing time to unit time). ρ = λ / μ |
| Average retention number | Lw | The average number of customers both waiting in the queue and currently being served. Lw = ρ / (1 − ρ) |
| Average waiting time | Tw | The average customer waiting time from a customer entering the queue until processing begins. Tw = Lw × Ts = Lw × (1 / μ) = ρ / [(1 − ρ) × μ] |
p. 54 · PDF 58
| Average processing time (Average response time) | Tq | The average processing time from a customer entering the queue until processing is complete. Tq = Tw + (1 / μ) |
|---|---|---|
| Average queue length | Lq | The average number of customers in the queue who are waiting to be served. Lq = Lw × ρ |
Source visual description
The original lottery-ticket example has arrival rate 4 customers per minute and service rate 5 customers per minute. Its displayed results are inter-arrival time 0.25 minutes/customer, service time 0.2 minutes/customer, usage 0.8, retention number 4 customers, waiting time 0.8 minutes, processing time 1.0 minutes and queue length 3.2 customers. Source variable names Lw, Tw, Tq and Lq and the original formulas are retained.
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Examples: At a particular lottery ticket stand, an average of 4 customers arrive every minute, and a lottery ticket can be sold (processed) to an average of 5 customers per minute. Calculate each numeric value of the queueing theory when this lottery ticket stand is assumed to follow an M/M/1 model. • Average arrival rate: (λ) = 4 customers / minute • Average inter-arrival time: (1/λ) = 1 minute / 4 customers = 0.25 minutes/customer • Average service rate: (μ) = 5 customers / minute • Average service time: (1/μ) = 1 minute / 5 customers = 0.2 minutes/customer • Usage rate: (ρ) = λ/μ = (4 customers/minute) / (5 customers/minute) = 0.8 • Average retention number: (Lw) = ρ / (1 − ρ) = 0.8 / (1 − 0.8) = 0.8 / 0.2 = 4 (customers) • Average waiting time: (Tw) = Lw × (1/μ) = 4 customers × 0.2 minutes/customer = 0.8 minutes • Average processing time: (Tq) = Tw + (1 / μ) = 0.8 minutes + 0.2 minutes = 1.0 minutes • Average queue length: (Lq) = Lw × ρ = 4 customers × 0.8 = 3.2 customers
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Original chapter exercises
Original chapter exercises →New FE Textbook Vol.2 · First Edition, April 2015