00094 �b88a889f5ea0de435f5ce77d7deced

A Very Simple Set of Process Control Rules

93

+1.5 to 3.0    pi    1

-1.5 to-3.0    p2    I+L2

> 13,01    I-P0-P1-P2    1

So, for Case II,

Li = the (additional) ARL given that Xi falls in [+1.5,3.0] = (1 +Lo)po+ l^xPi +(1 +L2)p2+ l^x(l ~Po^-Pi ■ P2) = I+L0P0 + L2P2

CASE HI: lst Observation falls in (-1.5 to -3.0)

[A2]

Valueof2nd observation (x₂)

-1.5 to 1.5 +1.5 to 3.0 -1.5 to-3.0 > 13.0 |

So, for Case III,

Prób. of 2nd    ARL starting

obs. (x₂)    from 2nd obs.

Po    1    +Lo

Pi    1    + L,

P2    1

I-P0-P1-P2    1

L2 = the (additional) ARL given that Xi falls in [-1.5,-3.0]

= (1 +U)po + (l +Li)pi + 1 P2+ 1 (1-P0-P1-P2)

= 1 +Lopo + L,pi

ARL = Average run length starting from lst observation (xi)

= (1 + Lo)po + (1 + L[)pi + (1 + L₂)p₂ +1 (1 - po-pi - p₂) = 1 + L0P0 + Lip! + L₂p₂ = Lq (from Al)

[A3]

[A4]

From Al & A2:

Lo-L] -L1P1 L, =Lo/(l+pi)

From A2 & A3:

Li -1^ = L₂p₂ - L] pi L2 =L, (1 +p,)/(l +pz)

[A5]