Waiting Line Management
Queuing systems: basic framework & key metrics
Customer
Population
Arrival
Exit
Queue
Service
1. Average utilization (% time server busy)
2. Average queuing time
3. Average queue length (# of customers in line)
4. Average system time (queuing + service)
5. Average # of customers in the system (in line + being
served)
Life without variability
Customer
Population
Arrival
Exit
Queue
Arrival stream:
Service time:
Average utilization
Average queuing time
Average queue length
Average system time
Average no. in system
Service
1 every 5 minutes 1 every 10 minutes
Exactly 5 minutes Exactly 5 minutes
100%
0
0
5
1
50%
0
0
5
0.5
Queuing Models…
Single queue, single server (M/M/1)
Arrival rate λ
Service rate μ
Assume
•
Time between arrivals is Exp(λ)
•
Time between services is Exp()
Queuing Models: Arrivals
T = time between arrivals
Assume T is an exponential random variable with rate l
Probability density function
for the exponential distribution:
Expected time between arrivals:
fT (t )  lelt , t  0
E[T ] 
1
l
Queuing Models: Arrivals
Time between arrivals is Exp(l)  Poisson arrivals at
rate l
If we want to know how many customers arrive in a
given time period, we can use the Poisson distribution.
N(T) = number of arrivals in T time units
(lT ) e
PN (T ) (n) 
n!
n
 lT
,n  0
PN(T)(n) is the probability that the number of arriving
customers in any period of length T is exactly n
Examples:
Customers arrive to a McDonalds according to a
Poisson process with rate 2 customers per minute.
1
1
 min
What is the expected time between arrivals? E[T ] 
l
2
What is the probability that exactly 3 customers arrive
in any 5 minute period?
(lT ) n e  lT
(2  5) 3 e 25
PN ( 5) (3) 

 0.0076
n!
3!
Queuing Models: Service
We assume that service time S is an exponential
random variable with rate 
Example: A bank teller can service customers at a rate
of 3 customers per minute.  = 3 customers/min
What is the expected service time?
1
1
E[ S ] 
 min
 3
Utilization for M/M/1…
How much of the capacity is being utilized?
Arrival rate
Utilization =
Service rate
l


Number of customers in the system
for the M/M/1 queue…
Ns = number of customers in system
PNs (n)  (1   ) , n  0
n
Average number of customers in the system
for the M/M/1 queue…
On average, how many customers are in the system
at any moment in time?
Ls = Average # of customers in the system
 0  PNs (0)  1 PNs (1)  2  PNs (2)  

l
 l
Metrics for the M/M/1 queue…
Utilization

l

Average # of customers in the system
Average # of customers in line
Ls 
l
 l
l2
Lq 
 (  l )
l
Average time a customer spends in line Wq 
 (  l )
Average time a customer spends in the system
1
Ws 
 l
Example …
Western National Bank is considering opening a drivethrough window for customer service. Management estimates
that customers will arrive at the rate of 15 per hour. The teller
who will staff the window can service customers at the rate
of one every three minutes.
Assuming exponential interarrival and service times,
calculate performance metrics of this queue.
Example (continued)
Because of limited space availability and a desire to provide
an acceptable level of service, the bank manager would like
reduce the probability that more than three cars are in the
system at any given time.
What is the current level of service?
What must be service rate at the teller, and what utilization
must be achieved, to ensure a 95% service level (probability
of having 3 or less cars in the system being 95%).
Queuing Models…
Single queue, multiple servers (M/M/s)
s = # of servers
Arrival rate λ
Service rate μ
(for each server)
Metrics for the M/M/s queue…
Utilization
l

s
Average # of customers in line
Lq
Average # of customers in the system
Average time a customer spends in line
(see table TN7.11)
l
Ls  Lq 

Wq 
Average time a customer spends in the system
Lq
l
Lq
1
Ws  
l 
Example …
Sharp Discounts Wholesale club has two service desks, one
at each entrance of the store. Customers arrive at each service
desk at an average of one every six minutes. The service rate
at each service desk is four minutes per customer.
a. What percentage of time is each service desk idle?
b. What is the probability that both desks are busy? Idle?
c. How many customers, on average, are waiting in line?
d. How much time does a customer spend at the service desk?
(waiting plus service time)
e. Should Sharp Discounts consolidate its two service desks?
The trade-off in waiting line management…
Cost of providing faster service vs. “cost” of waiting
Total cost
Cost
Cost of capacity
Cost of waiting
Capacity
Example …
In the service department at Glenn-Mark auto agency,
mechanics requiring parts for auto repair or service present
their request forms at the parts department counter. The parts
clerk fills a request while the mechanic waits. Mechanics
arrive according to a Poisson process with rate 40/h, and a
clerk can fill requests at the rate of 20/h. The costs for a parts
clerk is $6/h, and the cost for a mechanic is $12/h. Assuming
there are currently 3 parts clerks, would you add a fourth?