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Basic Queuing Insights

Basic Queuing Insights. Nico M. van Dijk “Why queuing never vanishes” European Journal of Operational Research 99 (1997) 463-476. Main Points. Simple questions such as “should separate queues be pooled?” do not have straightforward answers.

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Basic Queuing Insights

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  1. Basic Queuing Insights Nico M. van Dijk “Why queuing never vanishes” European Journal of Operational Research 99 (1997) 463-476

  2. Main Points • Simple questions such as “should separate queues be pooled?” do not have straightforward answers. • Continuous contention between customer and system perspectives. • Capacity and variation are the two most important factors in determining performance. • An inaccurate system model may be OK for design purposes.

  3. Customer vs. system perspectives Customer service System efficiency Maximize output per unit of capacity = X/ Y X = number of services actually completed per unit time Y = maximum number of services that could be completed per unit time • Minimize expected waiting time • Minimize probability of waiting more than a given length of time These objectives tend to conflict!

  4. Example of System/Customer Conflict 1. Single server with arrival rate l = 20/hr, service rate m = 30/hr 2. Two servers with total arrival rate 2l = 40/hr, combined service rate 2m = 60/hr Equivalent from system perspective but customers much prefer the second arrangement!

  5. Factors That Influence Delay • C = Capacity = maximum number of service completions per unit time (if all servers continuously busy) • s2 = Variation = variance of service time distribution • Also define: • A = arrival rate (customers per unit time) • R = average residual service time of a customer in service at an arbitrary instant. • S = expected service time • W = mean waiting time

  6. Single server with random arrivals Exponential service times Deterministic service times Fixed variation

  7. Supermarket case • Flexible capacity: number of cashiers depends on arrival rate • If customer cannot find a check-out with < 3 customers waiting, they get items free • Probability of this occurring with 5 check-outs is 1/3000 • Probability with 1 check-out is 1/5 • Guarantee costs 2% of sales but gross sales increased by 20%: guarantee of short waiting time.

  8. Pool or not? • Two types of customers: • Type 1: 50/hr arrive, fixed service time = 1 min. • Type 2: 5/hr arrive, fixed service time = 10 min. • Two dedicated servers: W1 = 2.5 min., W2 = 25 min., W = (10*2.5 + 1*25)/11 = 4.55 min. • Single queue for two servers: S = 1.82 min., s2 = 6.69 min2, W = 7.14 min.

  9. Postal office case • Short and long jobs • Banking and postal services • 5 servers with a single queue • Recommendation: • 2 counters for short banking jobs • 1 counter for long banking jobs • 2 counters for postal jobs • Some cross-traffic with priority

  10. General rules to lower total process times • Reduce variation in arrivals • Reduce variation in service times • Use capacity flexibly • Pool jobs of approximately equal durations • Specialize servers to jobs of different durations • Parallelize independent tasks • Combine dependent tasks • Prioritize

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