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.
• 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 l = 20/hr,
service rate m = 30/hr
2. Two servers with total arrival rate 2l =
40/hr, combined service rate 2m = 60/hr
Efficiency: 1 =
l
2l
= 66.7%,  2 =
= 66.7%
m
2m
Total delay: W1 = 6 minutes, W2 = 3.6 minutes
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)
• 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
Single server with random arrivals
Generic expressions
S = 1 C ,  = AS
Deterministic
service times
S =1 C
S
R=
2
S 
W=
2 1 
S 1s

2 2 S
A

W =R
=R
CA
1 
2
R=
Fixed
variation
scv =
s2
S2
1  scv
R=
2C
1  scv
A
W=
2 C  C  A
Exponential
service times
S =1 C
R=S
W =S

1 
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: 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.
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