Queues and Service Systems

Question 1

What are queues?

Answer 1

Queues are the result of the mismatch between demand and supply.

Queues can be real and virtual.
For example:
- Queuing up to pay money at the supermarket
- Waiting on the phone when customer service put us on hold.

Question 2

Queues when demand exceeds supply.

Answer 2

In this situation, queues will build up. So we want to find out:

1. What is the length of the queue
2. How long does the inventory need to wait to be processed
3. What is the average time inventory needs to wait to be processed.

Question 3

How to find the length of the queue?

Answer 3

Queue growth rate = Demand - Capacity

Length of queue at Time, T
= T * (Queue Growth Rate)
= T * (Demand - Capacity)

Question 4

Time to serve customers arriving at time, T.

Answer 4

Time to serve Qth person in the queue
= Q / Capacity, where Q is the queue length.

Time to serve customer arriving at time, T:
By using the formula to find queue length, we conclude that time to serve the person arrive at time T:
= T * (Demand / Capacity - 1)

Question 5

Average waiting time during a period T.

Answer 5

Average time to serve a unit
=1/2 * T * (Demand/Capacity - 1) , this T refers to the maximum period of the whole period.

2 factors that will determine average wait time:

1. Implied utilization (Demand/Capacity) -> This is a ratio
2. Duration of the period T.

Question 6

Managing Peak Demand (How to reduce demand or increase capacity)

Answer 6

How to reduce demand without reducing revenue?

1. Peak-load pricing:
   - Charging more during the time period with highest demand. (Compensation)
2. Off-peak discount
   - Offering a discount during a time period with low demand.

(BR)

Adjusting capacity

1. Pre-processing strategy:
   - Reducing the amount of work needed to process a customer during the peak time period by moving some of the work to an off-peak time.

This will result in less work per customer during peak time and the capacity to serve customers during that time is increased.

Question 7

Does demand always exceeds capacity?

Answer 7

No, this situation do not last forever. Hence managing peak demand is only temporary.

Question 8

Queues when demand is lower than supply and demand/service rates are variable - One server

Answer 8

Only one operator/person providing service.

Demand is lower than supply because we are using the flow rate/capacity. At one time, someone might still be waiting but we are looking from the perspective of the process -> What is the process capacity.

As long as the capacity is greater than the demand, we say that demand is lower than supply.

Question 9

What causes waiting time?

Answer 9

Variability leads to waiting time and inventory.

Question 10

Variability: Where does it come from?

Answer 10

Inputs:

• Unpredicted volume swings
• Random arrivals
• Incoming quality
• Product mix

Tasks:

• Inherent variation
• Lack of SOP
• Quality (Scrap/rework)

Resources:

• Breakdowns /Maintenance
• Operator absence
• Set-up times

Routes:

• Variable routing
• Dedicated machines

There are countless reasons that leads to variability.

Question 11

The arrival and service processes

Answer 11

What is arrival process:
- The flow of customers arriving to the system

What is service process:
- The flow of customers when they are being served.

What is average inter-arrival time:
- The average time between customers arrivals to a system (a)

What is average processing time:
- The average time a customer spends with a server (p) -> Service time

Question 12

Measuring of variability: First source of variability -> The arrival process.

Answer 12

Coefficient of Variation = Standard Deviation / Mean

Question 13

Measuring of variability: Second source of variability -> The service times

Answer 13

Coefficient of Variation = Standard Deviation of activity time / Average activity time

For exponential distribution, CV = 1., where CV = coefficient of variation.

Question 14

What is the flow rate, flow time and assumptions made of a queuing model with a single server?

Answer 14

Flow rate, R = 1 / a

Capacity = 1 / p

Inventory in process: Ip

Inventory in queue: Iq

Flow time = Tq + p (Waiting time + Processing time)

Assumptions made:

1. All incoming customers only join a single queue.
2. When customers arrive, they will wait till the server is completed. They will not just leave because the queue is too long. This means that they will stick through and not leave halfway. (Patient customer)
3. The average inter arrival time is greater than the average processing time. There is enough capacity to service the incoming demand.

Question 15

Utilization

Answer 15

Utilization
= Flow rate / Capacity
= (1 /a) / (1 / p)
= p / a

• p being less than a is assumed in this model.
• This means that the time to process a customer is shorter than the time between customer arrivals - and - the capacity of the server, 1/p, is greater than the demand rate.

Question 16

Time in queue

Answer 16

Time in queue
= Service Time x [utilization / (1 - utilization)] x [(CVa squared + CVp squared) / 2]
= service time(p) x utilization factor x variability factor

(BR)

• The larger the service time (p), the lower the capacity (1/p).
• As utilization increase, the utilization factor will increase, hence increasing the time in the queue.

(BR)

Total flow time in a system = Time in queue (Tq) + Time in service (processing time, p)

Question 17

Prediction of the number of customers waiting and in service

Answer 17

Iq = average number of customers waiting. 
Ip = average number of customers in service. 

We can apply little’s law because these are inventories.

Iq
= R * Tq, where R is the flow rate
= (1/a) * Tq
= Tq / a

Ip
= R * p, where R is the flow rate
= (1/a) * p
= p / a 
= Utilization

Question 18

Queues when demand is lower than supply and demand/service rates are variable - Multiple Servers

Answer 18

• m identical servers. Each of these servers requires a processing time of p.
• average processing time, p
• waiting time, Tq
• processing time, p
• flow time = Tq + p
• All customers join a single queue and it will be a first come first serve basis.
• Once the customers arrive, they will wait till the service is completed.
• p is the average service time of each of the m servers.
• average inter arrival of customers, a.
• arrival rate = 1 / a. This is also the flow rate in the system.
• CV of a and CV of p.
• capacity of the system is assumed to be greater than the flow rate.
• each of these servers have a capacity of 1/p.
• total capacity = 1 upon p and this should be greater than the arrival rate.

Question 19

Predicting the waiting time in Queue, Tq, waiting time in service and total time in the system (multiple servers)

Question 20

Utilization for multiple parallel resources

Answer 20

Utilization:
= Flow rate / Capacity
= (1/interarrival time) / (m x (1/activity time)
= (1/a) / (m/p)
= p / (a x m) -> This needs to be less than 1 (To show that the demand is less than the capacity)

Assuming that the demand is less than the capacity.

Question 21

Time in queue (multiple servers)

Answer 21

Time in queue
= Activity time/service time factor X Utilization factor X Variability Factor

• Higher activity time means higher time in queue.
• As utilization increases, the time in queue also increases.
• As variability increases, the time in queue also increases.

Question 22

What is an unstable queue?

Answer 22

A queuing system in which the demand rate > capacity.

Queue may start increasing and increasing because demand> capacity.

Queue may be out of control in the long run.

Question 23

What is a stable queue?

Answer 23

A queuing system in which the demand rate < capacity.

Queue will be kept in control because demand rate < capacity.

Question 24

Factors affecting waiting time in queue for multiple server systems