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

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.

Customer vs. system perspectives Customer service Minimize expected waiting time Minimize probability of waiting more than a given length of time 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 These objectives tend to conflict!

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

Factors That Influence Delay C = Capacity = maximum number of service completions per unit time (if all servers continuously busy)  2 = 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

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

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.

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: W 1 = 2.5 min., W 2 = 25 min., W = (10* *25)/11 = 4.55 min. Single queue for two servers: S = 1.82 min.,  2 = 6.69 min 2, W = 7.14 min.

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

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