1. Waiting Line Management

2. Queuing systems: basic framework & key metrics
Customer
Population
Arrival
Queue
Service
Exit
Average utilization (% time server busy)
Average queuing time
Average queue length (# of customers in line)
Average system time (queuing + service)
Average # of customers in the system (in line + being served)

3. Life without variability
Customer
Population
Exit
Arrival
Queue
Service
Arrival stream: 1 every 5 minutes
Service time: Exactly 5 minutes
1 every 10 minutes
Exactly 5 minutes
Average utilization
Average queuing time
Average queue length
Average system time
Average no. in system
100% 5 1
50% 5
0.5

4. Single queue, single server (M/M/1)
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()

5. 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:

6. Queuing Models: Arrivals
Time between arrivals is Exp()  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
PN(T)(n) is the probability that the number of arriving customers in any period of length T is exactly n

7. Examples:
Customers arrive to a McDonalds according to a Poisson process with rate 2 customers per minute.
What is the expected time between arrivals?
What is the probability that exactly 3 customers arrive in any 5 minute period?

8. 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?

9. Utilization for M/M/1… How much of the capacity is being utilized?
Arrival rate
Service rate

10. Number of customers in the system for the M/M/1 queue…
Ns = number of customers in system

11. 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

12. Metrics for the M/M/1 queue…
Utilization
Average # of customers in the system
Average # of customers in line
Average time a customer spends in line
Average time a customer spends in the system

13. Example … Western National Bank is considering opening a drive-
through 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.

14. 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%).

15. Single queue, multiple servers (M/M/s)
Queuing Models…
Single queue, multiple servers (M/M/s)
s = # of servers
Arrival rate λ
Service rate μ
(for each server)

16. Metrics for the M/M/s queue…
Utilization
Average # of customers in line
(see table TN7.11)
Average # of customers in the system
Average time a customer spends in line
Average time a customer spends in the system

17. 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.
What percentage of time is each service desk idle?
What is the probability that both desks are busy? Idle?
How many customers, on average, are waiting in line?
How much time does a customer spend at the service desk?
(waiting plus service time)
e. Should Sharp Discounts consolidate its two service desks?

18. Cost of providing faster service vs. “cost” of waiting
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

19. 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?