OMIS 2010: Queue / Service Design (Week 10)

What is a queue?

process flow with input &/ service variability

What explains why a queue has a buffer?

variability

What are the queue model uses?

• describe variable system’s unit behaviour

• determine service level

• evaluate alternate service

What are the queue parts?

• arrival

• wait

• service

• system notation

• performance measure

What are the arrival aspects?

• population

• behaviour

• distribution

What are the wait aspects?

• max size

• discipline

• line number

What are the service aspects?

• server number

• distribution

What are the queue performance measures?

• time

  • queue

  • system

• count / length

• other

  • utilization

  • wait probability

What are the arrival’s population attributes?

• infinite

>

• finite

What are the arrival’s distribution attributes?

• scheduled

• random

What is the arrival’s random distribution?

exponential

• density

• mean // standard deviation

• variance

What is the arrival’s random distribution density formula?

$$f\left(t\right)=\lambda e^{-\lambda t}$$

What is the λ arrival’s random distribution notation?

arrival rate

What is the arrival’s random distribution mean // standard deviation formula?

$$\frac{1}{\lambda}$$

What is the arrival’s random distribution variance formula?

$$\frac{1}{\lambda^2}$$

What is the memoryless property?

next arrival does NOT depend on last arrival → poisson distribution

What is the N_t arrival’s random distribution notation?

arrival number before t

What is the occurrence number = n probability formula?

$$P\left(N_{t}=n\right)=\frac{\left(\left(\lambda t\right)^{n}e^{-\lambda t}\right)}{n!}$$

What is the first occurrence wait time formula?

$$P\left(N_{t}=0\right)=e^{-\lambda t}$$

What are the arrival’s behaviour attributes?

• balk

• renege

• jockey

• patient

What is balking?

customers arrive & leave after inspecting queue

• observable queue

• NOT enter wait area

• if NO buffer, balk

What is reneging?

customers join & leave queue

• NOT needed observable queue

• impatient

What is jockeying?

customers join queue & switch lines

What is the wait’s max size attribute?

NO buffer limit BUT physical limit

What are the wait’s discipline attributes?

• FIFO // FCFS

• LIFO

• random

• priority

What are the wait’s line number attributes?

• 1 line

>

• 1/server

What are the service’s server number attributes?

• more servers → more timely service / happier customer BUT cost

• cross-train server

What are the service’s time distribution attributes?

• deterministic

• random

What is the service’s time distribution?

exponential

• density

• mean

• variance

What is the service’s time distribution density formula?

$$g\left(t\right)=\mu e^{-\mu t}$$

What is the service’s time distribution mean // standard deviation formula?

$$\frac{1}{\mu}$$

What is the service’s time distribution variance formula?

$$\frac{1}{\mu^2}$$

What is the queue notation?

Kendall

• A

• S

• c // s

• K

• N

• D

What is the A Kendall notation?

arrival time distribution

What is the S Kendall notation?

service time distribution

What is the c Kendall notation?

server number

What are the A parameters?

• M

  • memoryless IF exponential

• G

  • general

• D

  • deterministic

What are the S parameters?

• M

  • memoryless

• G

  • general

• D

  • deterministic

What are the c // s parameters?

• 1

• s

What are some Kendall notation examples?

• M/M/1

• M/M/s

• M/G/1

• M/D/1

What does M/M/1 mean?

• poisson arrival

• poisson service

• 1 server

What does M/M/s mean?

• poisson arrival

• poisson service

• 1/server

What does M/G/1 mean?

general service

What does M/D/1 mean?

constant service

What are the predictable variation performance measures?

• average

• utilization

What are the unpredictable variation performance measures?

• average

  • wait / service time & size

• probability

  • idle

  • system number

• utilization

What are the Little’s Flow equations?

average system / queue time & average system / queue units relation

$$L=\lambda\cdot W$$

$$L_{q}=\lambda\cdot W_{q}$$

$$W=\frac{L}{\lambda}=W_{q}+\frac{1}{\mu}$$

$$W_{q}=\frac{L_{q}}{\lambda}$$

What increases throughput if the arrival rate is variable?

• increase staff

• decrease service time

• buffer

What are the operation change characteristics?

• arrival rate

  • smooth demand by price promotion

• server number / type

  • dedicate server to task

• server efficiency

  • add resource / tech

• line number

  • dedicate server to transaction

• phase number

• priority

What are the M/M/1 characteristics?

• homogeneous / infinite / patient population & 1 queue

• FIFO

• 1 server

• exponential arrival & service time distribution

What if W_q is long?

• hire cashier assistant → increase service rate

• hire cashier → decrease service rate

What is the L_q M/M/1 shortcut formula?

$$Lq=\rho L$$

What is the W_q M/M/1 shortcut formula?

$$W_{q}=pW$$

What is the P_n M/M/1 shortcut formula?

$$P_{n}=\left(1-\rho\right)\rho^{n}$$

What are the M/M/s characteristics?

• homogeneous / infinite / patient population & 1 queue

• FIFO

• multiple servers

• exponential arrival & service time distribution

What is the M/G/1 use?

NOT all service time → exponential distribution