Science Talk-100111-陸行

誰想計算等候時間？ Computing Waiting Time! Who Cares? 政治大學應用數學系陸行 Hsing Luh Department of Mathematical Sciences

貓空纜車假日試辦「進站時間號碼牌」管制
「進站時間號碼牌」管制措施是以 10 分鐘為單位，每單位設定運送遊客 200 人。民眾只要告知工作人員欲搭乘人數，就可領到一張號碼牌，牌上載明進站時段，在該時段回到入口處就可進站搭車；逾 10 分鐘以上須重抽牌。

多線排隊

單線排隊多人服務

等候時間的計算與應用
單線、多線排隊
如何描述等候時間

如何描述等候時間
平均等候時間
等候時間的變異數
等候時間的機率分配函數
Agner Krarup Erlang (1878-1929)

到達時間是隨機變數

服務時間是隨機變數

等候時間是隨機變數

An M/M/1 Queueing system
P (exact 1 arrival in [t, t+  t])=  t
P (no arrivals in [t, t+  t])= 1-  t
P (1 service completion in [t, t+  t])=  t
P (no service completion in [t, t+  t])= 1-  t
P n (t) = P (# of customers is n at time t)
Courtesy to Internet Information

狀態平衡方程式 Courtesy to Internet Information

n-1 n n+1         Courtesy to Internet Information 狀態平衡方程式

長時間的觀察值 Courtesy to Internet Information n-1 n n+1        

基本假設條件
Kirchov’s current conservation law
Flow in = Flow out
P 0 + P 1 +…+ P N =1

區域狀態平衡方程式
 P 0 =  P 1
 P 1 =  P 2
 P 2 =  P 3
… ..
 P n-1 =  P n
Courtesy to Internet Information 0 1 2 3 n-1 n n+1                

機率分配
P 1 = (  /  ) P 0
P 2 = (  /  ) P 1
… ..
P n = (  /  ) P n-1

Little’s Law
The mean number of customer in a queueing system
L=  W W is the mean waiting time

Example of Little Law
The average waiting time for M/M/1 queue
Little Law also holds if we consider only the queue and not the server.

State dependent M/M/1 Queueing Systems 0 1 2 3 n-1 n n+1  (0)  (1)  (2)  (3)  (n-2)  (n-1)  (n)  (n+1)  (1)  (2)  (3)  (4)  (n-1)  (n)  (n+1)  (n+2) Courtesy to Internet Information

Performance Measure
Mean Throughput of the queueing system
Mean rate at which customers pass through
Mean number of customers in the queueing system
Mean time delay (using Little’s Law)
Utilization
U=1-p 0

Example 1 Suppose interarrival time is 12 min and service time is 10 minutes per job. . M/M/1:

M/M/1/N Queueing System The Finite Buffer Case
Blocking probability
Courtesy to Internet Information  0 1 2 3 N-1 N           

Example 2 Suppose interarrival time is 12 min and service time is 10 minutes per job. . M/M/2:

比比看 Two M/M/1 One M/M/2

平均等候時間平均服務率 M/M/1 平均到達率 P{idle time} E(waiting time) × 2

平均等候時間平均服務率 M/M/2 平均到達率 P{idle time} } E(waiting time}

Example 3 Suppose interarrival time is 12 min and service time is 10 minutes per job. . Two M/M/1: One M/M/2: × 2

M/M/2 與 M/M/1 的比較 < < < × 2 × 2 in queue) <

等候時間的機率分配函數

佇列的等候機率

時間延誤問題系統模擬

系統模擬 (1)

系統模擬 (2)

系統模擬 (3)

交通號誌

匝道管制

通訊問題

通訊服務

An optimization model Minimize Penalty made by unsatisfied traffic aggregates Subject to Fixed resource (Limited Short Path bandwidth) Traffic aggregate QDF budget must be satisfied Allocate resource to traffic aggregates Trading off QDF with bandwidth in each Short Path 最佳化模型

Introduction
They tried to get the optimal solution of the model as follows:
.
.

Utility Functions
An utility function of (the bandwidth of each class ):
where aspiration and reservation levels,
and , respectively, and .
,

Utility Functions
The graph of an utility function :

A Network Optimization Model

A Network Optimization Model
is a set satisfying constraints of the Model .

Network Constraints
To determine whether link e is chosen we define the binary decision variable:
For each connection j of class i, we denote the routing path connecting the source node o and destination node d by .

Network Constraints
It shows that every connection in the same class uses the same bandwidth and has the same bandwidth requirement.

Network Constraints
We have the budget constraint on the network:
Since every connection has the maximal end-to-end delay constraint.

Network Constraints
Flow constraints:

An Illustrative Example
Network Topology for an Illustrative Example

:

QoS: Quality of Service.

Numerical Results

參考文獻 Hsing Luh, Chia-Hung Wang, and Chih-Chin Liang, “Measuring the Cost of Links of a Disaster-Avoided Content-Delivering Service within a Large-scaled Company,” Proc. of the Fourth International Conference on Queueing Theory and Network Applications (QTNA2009), Fusionopolis, Singapore, July 29-31, 2009.
Hsing Luh and Chia-Hung Wang, “A Fast Branch and Bound Based Algorithm for Bandwidth Allocation and QoS Routing on Class-based IP Networks,” Proc. of the First World Congress on Global Optimization in Engineering & Science (WCGO2009), Hunan, China, June 1-5, 2009.
Hsing Luh and Chia-Hung Wang, “Measuring Average Revenue with Bandwidth and Traffic Demand on Communication Networks,” Proc. of International Conference on Computational Intelligence and Software Engineering (CiSE 2009), Wuhan, China, December 11-13, 2009.
Hsing Luh, Chih-Chin Liang, Chia-Hung Wang, Ping-Yu Hsu, and Wuyi Yue, “Proactive Peer-to-Peer Traffic Control when Delivering Large Amounts of Content within a Large-Scale Organization,” Proc. of the Ninth International Conference on Next Generation Wired/Wireless Networking (NEW2AN 2009), St. Petersburg, Russia, September 15-18, 2009.
Ji-hong Li, Nai-shuo Tian, Zhe George Zhang, Hsing Paul Luh, “Analysis of the M/G/1 queue with exponentially working vacations —A matrix analytic approach.” Queueing Systems, Vol. 61, 2009, 139-166. DOI 10.1007/s11134-008-9103-8.
Chia-Hung Wang and Hsing Luh, “A Utility Maximization Scheme for End-to-End QoS Routing in Communication Networks with Heterogeneous Service Classes,” 2008 IEEE International Conference on Service Operations and Logistics, and Informatics, Beijing, Oct, 2008.

作業研究
所謂「運籌」，就是「調度」，就是將有限資源做有效及最佳調度與配置的意思。
例如：一個企業管理者必須將原料、半成品、成品、存貨、勞動力、資金、時間，做最佳的調度與組合，期使成本降到最低，利潤增到最大。資源最佳值有效調度配置目標

運籌科學與相關領域 OR and Related Fields
作業研究
運籌學
Operations Research
數學系、應用數學系 Department of Mathematics, Department of Mathematical Sciences
工業工程系、資訊科學系 Department of Industrial Engineering, Department of Computer Science
工商管理系、統計系 Department of Business Administration, Department of Statistics
線性規劃、非線性規劃、動態規劃 Linear Programming, Nonlinear Programming, Dynamic Programming
等候理論、隨機規劃、系統模擬 Queueing Theory, Stochastic Programming, System Simulation

運 OR 籌資訊 Information 管理 Management 科學 Science

By Bertrand Mareschal 1. Introduction History. Methodology. Models. 2. Linear programming Applications. Simplex algorithm. Sensitivity analysis. Integer programming. 3. Network analysis Transportation problems. Maximum flow problems. CPM and PERT – Project management. 4. Inventory management Deterministic EOQ models. Probabilistic models. Production management 5. Multicriteria decision aid Methods. Multicriteria decision support systems. Group decision systems. Applications.

模式與方法 Models and Methodologies Models deterministic probabilistic -------------------------------------------------- optimization / OR OR solutions Methodologies simulation --------------------------------------------------    

By Bertrand Mareschal
Some of the most widely used techniques :
Linear programming :
Optimization under constraints
Production planning, scheduling, financial planning,
Simulation methods :
Simulate complex systems
Queueing
PERT/CPM :
Complex projects management
Network models :
Transportation problems
Inventory management :
Reduce inventory levels and costs
JIT production management

Statistical analysis :
Summarize and analyze data.
Multicriteria decision aid :
Identify best compromise solutions
When conflicting objectives are considered
Group decision aid :
Achieve consensus efficiently
Dynamic programming :
Repetitive (w.r.t. time) decision problems.
Reliability theory :
Reliability of equipments.

供應鏈 Supply Chain Management The Goal: A Process of Ongoing Improvement by Eliyahu M. Goldratt

推銷員問題 Traveling Salesman Problem TSP http://www.ing.unlp.edu.ar/cetad/mos/TSPBIB_home.html O(2^n) 2^50 ~10^15 3*10^7* 10^6 ~ 30 years

BIOINFORMATICS 生物資訊 http:// motif.stanford.edu/index.html

CRYTOGRAPHY 密碼學 Elliptic Curve http:// www.certicom.ca

資源 Resource 最佳值 Optimum 有效調度 Adjustment 配置 Allocation 目標 Objective 動態系統

最佳解改進最佳解可行解

No interaction allowed 2 1 Individual sampling model (ISM): Simulated patient-level Markov model (SPLMM) (variations as in quadrant below for patient level models with interaction) Simulated Markov model (SMM) Markov model (evaluated deterministically) Timed Individual sampling model (ISM): Simulated patient-level decision tree (SPLDT) Simulated decision tree (SDT) Decision tree rollback Untimed Non-Markovian, discrete-state, individuals Markovian, discrete state, individuals Markovian, discrete state, stochastic Expected value, continuous state, deterministic Individual level Cohort/aggregate level/counts D C B A

Discrete event simulation (CT, DES) Discrete individual simulation (DT, DES) Interaction allowed 4 3 Continuous time Individual event history model (CT, IEH) Continuous time Markov chain model (CTMC) System dynamics (ordinary differential equations, ODE) Continu-ous time Discrete-time individual event history model (DT, IEH) Discrete time Markov chain model (DTMC) System dynamics (finite difference equations, FDE) Discrete time Non-Markovian, discrete-state, individuals Markovian, discrete state, individuals Markovian, discrete state, stochastic Expected value, continuous state, deterministic Individual level Cohort/aggregate level/counts D C B A

Queueing Theory -- Establishing policies for sending in resupply aircraft to small airfields with maximum on ground (MOG) limits. For
example, the airfield at Haiti had a MOG of 3 C-5s.
Decision Analysis -- A recent study was completed that looked at
methods for considering the counter proliferation effects of weapons
of mass destruction when evaluating new systems.
Response Surface Methodology -- Using multiple runs of a large
theater-level model, Thunder, to build a response surface allowing
quick analysis of effects of changes in air apportionment.

謝謝大家

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