Graduate Lectures and Problems in QualityControl and Engineering Statistics:Theory and MethodsTo AccompanyStatistical Quality Assurance Methods for EngineersbyVardeman and JobeStephen B. VardemanV2.0: January 2001c° Stephen Vardeman 2001. Permission to copy for educationalpurposes granted by the author, subject to the requirement thatthis title page be a¢xed to each copy (full or partial) produced.Chapter 2Process MonitoringChapters 3 and 4 of V&J discuss methods for process monitoring. The keyconcept there regarding the probabilistic description of monitoring schemes isthe run length idea introduced on page 91 and speci…cally in display (3.44).Theory for describing run lengths is given in V&J only for the very simplest caseof geometrically distributed T . This chapter presents some more general toolsfor the analysis/comparison of run length distributions of monitoring schemes,namely discrete time …nite state Markov chains and recursions expressed interms of integral (and di¤erence) equations.2.1 Some Theory for Stationary Discrete TimeFinite State Markov Chains With a SingleAbsorbing StateThese are probability models for random systems that at times t = 1; 2; 3 : : :can be in one of a …nite number of statesS1; S2; : : : ; Sm; Sm+1:The “Markov” assumption is that the conditional distribution of where thesystem is at time t + 1 given the entire history of where it has been up throughtime t only depends upon where it is at time t. (In colloquial terms: Theconditional distribution of where I’ll be tomorrow given where I am and how I gothere depends only on where I am, not on how I got here.) So called “stationary”Markov Chain (MC) models employ the assumption that movement betweenstates from any time t to time t + 1 is governed by a (single) matrix of (one-step) “transition probabilities” (that is independent of t)P(m+1)£(m+1)= (pij)wherepij= P [system is in Sjat time t + 1 j system is in Siat time t] :2122 CHAPTER 2. PROCESS MONITORINGS SS1 23.1 .05.8.051.0.1.9Figure 2.1: Schematic for a MC with Transition Matrix (2.1)As a simple example of this, consider the transition matrixP3£3:=0@:8 :1 :1:9 :05 :050 0 11A: (2.1)Figure 2.1 is a useful schematic representation of this model.The Markov Chain represented by Figure 2.1 has an interesting property.That is, while it is possible to move back and forth between states 1 and 2,once the system enters state 3, it is “stuck” there. The standard jargon for thisproperty is to say that S3is an absorbing state. (In general, if pii= 1, Siiscalled an absorbing state.)Of particular interest in applications of MCs to the description of processmonitoring schemes are chains with a single absorbing state, say Sm+1, where itis possible to move (at least eventually) from any other state to the absorbingstate. One thing that makes these chains so useful is that it is very easy towrite down a matrix formula for a vector giving the mean number of transitionsrequired to reach Sm+1from any of the other states. That is, withLi= the mean number of transitions required to move from Sito Sm+1;Lm£[email protected]; P(m+1)£(m+1)=0@Rm£mrm£101£m11£11A; and 1m£[email protected] is the case thatL = (I ¡ R)¡11 : (2.2)2.1. SOME THEORY FOR STATIONARY DISCRETE TIME FINITE STATE MARKOV CHAINS WITH A SINTo argue that display (2.2) is correct, note that the following system of mequations “clearly” holds:L1= (1 + L1)p11+ (1 + L2)p12+ ¢ ¢ ¢ + (1 + Lm)p1m+ 1 ¢ p1;m+1L2= (1 + L1)p21+ (1 + L2)p22+ ¢ ¢ ¢ + (1 + Lm)p2m+ 1 ¢ p2;m+1...Lm= (1 + L1)pm1+ (1 + L2)pm2+ ¢ ¢ ¢ + (1 + Lm)pmm+ 1 ¢ pm;m+1:But this set is equivalent to the setL1= 1 + p11L1+ p12L2+ ¢ ¢ ¢ + p1mLmL2= 1 + p21L1+ p22L2+ ¢ ¢ ¢ + p2mLm...Lm= 1 + pm1L1+ pm2L2+ ¢ ¢ ¢ + pmmLmand in matrix notation, this second set of equations isL = 1 + RL : (2.3)SoL ¡ RL = 1 ;i.e.(I ¡ R)L = 1 :Under the conditions of the present discussion it is the case that (I ¡ R) isguaranteed to be nonsingular, so that multiplying both sides of this matrixequation by the inverse of (I ¡ R) one …nally has equation (2.2).For the simple 3-state example with transition matrix (2.1) it is easy enoughto verify that withR =µ:8 :1:9 :05¶one has(I ¡ R)¡11 =µ10:511¶:That is, the mean number of transitions required for absorption (into S3) fromS1is 10:5 while the mean number required from S2is 11:0.When one is working with numerical values in P and thus wants numericalvalues in L, the matrix formula (2.2) is most convenient for use with numericalanalysis software. When, on the other hand, one has some algebraic expressionsfor the pijand wants algebraic expressions for the Li, it is usually most e¤ectiveto write out the system of equations represented by display (2.3) and to try andsee some slick way of solving for an Liof interest.It is also worth noting that while the discussion in this section has centeredon the computation of mean times to absorption, other properties of “time toabsorption” variables can be derived and expressed in matrix notation. Forexample, Problem 2.22 shows that it is fairly easy to …nd the variance (orstandard deviation) of time to absorption variables.24 CHAPTER 2. PROCESS MONITORING2.2 Some Applications of Markov Chains to theAnalysis of Process Monitoring SchemesWhen the “current condition” of a process monitoring scheme can be thoughtof as discrete random variable (with a …nite number of possible values), because1. the variables Q1; Q2; ::. fed into it are intrinsically discrete (for examplerepresenting counts) and are therefore naturally modeled using a discreteprobability distribution (and the calculations prescribed by the schemeproduce only a …xed number of possible outcomes),2. “discretization” of the Q’s has taken place as a part of the developmentof the monitoring scheme (as, for example, in the “zone test” schemesoutlined in Tables 3.5 through 3.7 of V&J), or3. one approximates continuous distributions for Q’s and/or states of thescheme with a “…nely-discretized” version in order to approximate exact(continuous) run length properties,one can often apply the material of the previous section to the prediction ofscheme behavior. (This is possible when the evolution of the monitoring schemecan be thought of in terms of movement between “states” where the conditionaldistribution of the next “state” depends only on a distribution for the next Qwhich itself depends only on the current “state” of the