Queuing Systems: basic elements

Arrivals ServiceWaitingline

Exit

Processingorder

System

Queuing Systems: basic elements

Multiple channel

Multiple phase

Queuing Systems: multiple phases

Modeling with Queuing Theory

System Characteristics– Population source: finite, infinite– No. of servers– Arrival and service patterns: e.g. exponential

distribution for inter-arrival time– Queue discipline: e.g. first-come-first-serve

Measuring Performance

Performance Measurement:– System utilization– Average no. of customers: in line and in system– Average waiting time: in line and in system

e.g. infinite source, single server, exponential inter-arrival and service times, first-come-first-serve: (see handout)

Optimum

Cost of service capacity

Cost of service capacity

Cost of customerswaiting

Cost of customerswaiting

Total costTotal cost

Co

st

Service capacity

Totalcost

Customerwaiting cost

Capacitycost= +

Basic Tradeoff

System Utilization

Av

era

ge

nu

mb

er o

n

tim

e w

ait

ing

in li

ne

0 100%

Basic Tradeoff (cont.)

Applying Queuing Theory

In Process Design:– Describe the process and establish a model– Collect data on incoming and service patterns– Find formulas and/or tables, software to calculate

performance measures– Use performance measures to guide process design

decisions

Applying Queuing Theory

In Operations:– Monitor performance measures– Use performance measures to guide process

improvement and operations decisions

Statistical Process Control

Emphasis on the process instead of the product/material

Focus on “prevention”

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

UCL

LCL

Sample number

Mean

Out ofcontrol

Normal variationdue to chance

Abnormal variationdue to assignable sources

Abnormal variationdue to assignable sources

Control Chart

Sample number

UCL

LCL

1 2 3 4

In-Control: random only

Control Charts for Variables

Mean Chart: measuring sample means Range Chart: measuring sample ranges

i.e. max-min

UCL

LCLUCL

LCL

R-chart

x-Chart Detects shift

Does notdetect shift

process mean is shifting upward

SamplingDistribution

Out-of-Control: assignable & randomshifted mean

UCL

LCL

LCL

R-chart Reveals increase

x-Chart

UCL

Does notreveal increase

(process variability is increasing)SamplingDistribution

Out-of-Control: assignable & randomincreased variability

Mean

LCL UCL

/2 /2

Probabilityof Type I error

Type I Error:

MeanLCL UCL

Type II Error:

In-Control Out-of-Control

p-Chart - Control chart used to monitor the proportion of defectives in a process

c-Chart - Control chart used to monitor the number of defects per unit

Control Charts for Attributes

Counting Above/Below Median Runs (7 runs)

Counting Up/Down Runs (8 runs)

U U D U D U D U U D

B A A B A B B B A A B

Counting RunsFigure 10-11

Figure 10-12

LowerSpecification

UpperSpecification

Process variability matches specifications

LowerSpecification

UpperSpecification

Process variability well within specifications

LowerSpecification

UpperSpecification

Process variability exceeds specifications

Process Capability

Processmean

Lowerspecification

Upperspecification

1350 ppm 1350 ppm

1.7 ppm 1.7 ppm

+/- 3 Sigma

+/- 6 Sigma

Process Capability: 3-sigma & 6-sigma

Input/Output Analysis

Change in inventory = Input - Output Average throughput time is proportional to the

level of inventory.

Input flow of materials

Inventory level

Scrap flow

Output flow of materials

Flow and InventoryFlow and Inventory

Figure 11.1

MRP

A general framework for MRP Inputs: Bill of Materials, Inventory Files and

Master Production Schedule MRP Processing

A General Framework of MRP

Aggregate Plan

Master ProductionSchedule

MRP

Capacity RequirementsPlanning

Production Scheduling

Master Production ScheduleMaster Production Schedule

Week 1 2 3 4 5 6 7 8

M1 23 23 23 23

M2 10 10 10

Bill of MaterialsBill of MaterialsC (1)Seat

subassembly

H (1)Seat

frame

I (1)Seat

cushion

J (4)Seat-frame

boardsFigure 15.10

Inventory Files

On-Hand Open Orders Lead Times Vendor Information Quality records, etc.

MRP ExplosionMRP Explosion

Item: Seat subassemblyLot size: 230 units

Lead time: 2 weeks

Gross requirements 150150

1

230230

117117

2 3

120120

4 5

150150

6

120120

7 8

Scheduled receipts

Projected on-hand inventory

Planned receipts

Planned order releases

37

Week

117117 117117

0 00 0

00 00 000 00 0

227 227 77 187 187

230230

230230

Figure 15.11

Item: Seat subassemblyLot size: 230 units

Lead time: 2 weeks


1 2 3

120120

4 5

150150

6

120120

7 8

Planned receipts


Week

0 00 0

230

230

230

230

Item: Seat framesLot size: 300 units

Lead time: 1 week


1

00

2 3

00

4 5 6 7 8

Scheduled receipts


Planned receipts


40

Week

230 2300

00 00 00300 00 0

Item: Seat cushionLot size: L4L

Lead time: 1 week


1

00

2 3

00

4 5 6 7 8

Scheduled receipts


Planned receipts


0

Week

230 2300

00 00 000 00 0

Usage quantity: 1 Usage quantity: 1

MRP ExplosionMRP Explosion

Figure 15.11

Issues in MRP

Two basic concepts:– Net requirements– Lead time offset

Lot size Safety stock/Safety lead time Inventory records Validity of the schedules

JIT and Inventory Management

Inventory as delay in work flow Why inventory?

– Dealing with fluctuations in demand– Dealing with uncertainty– Reducing transaction costs– Taking advantage of quantity discount– Hedging against inflation, etc.


Inventory costs:– Holding cost– Long response time– Low flexibility– Slow feedback in the system


The objective of JIT: – General: reduce waste– Specific: avoid making or delivering parts before

they are needed

Strategy:– very short time window– mixed models– very small lot sizes.


Prerequisites:– Reduce set up time drastically– Keep a very smooth production process

Core Components:– Demand driven scheduling: the Kanban system– Elimination of buffer stock


Core Components: (cont.)– Process Design:

Setup time reduction Manufacturing cells Limited work in process

– Quality Improvement

Documents

Queuing Systems: basic elements