Question

The Markov Model on Excel tab depicts the population of Warrant Officers in the Marine Corps. A Warrant Officer is individual who has spent considerable enlisted time acquiring a high level of expertise in an occupational specialty and is awarded an officer’s commission to fill various leadership roles in that community. There are five grades (WO-CWO5). In the system we model here, there are three absorbing states. Two are self-explanatory (resign, retire) but the other is transition/promotion to Limited Duty Officer (LDO). This just means that they leave this particular system and become Lieutenants or Captains with certain restrictions on the billets they can hold. The matrix indicates the annual transition probabilities between states.

EXCEL tab is given as:

	WO	CWO2	CWO3	CWO4	CWO5	LDO	Retire	Resign
WO	0.68	0.2	0	0	0	0	0	0.12
CWO2	0	0.65	0.23	0	0	0.08	0.01	0.03
CWO3	0	0	0.64	0.26	0	0.07	0.02	0.01
CWO4	0	0	0	0.7	0.2	0.02	0.08	0
CWO5	0	0	0	0	0.87	0	0.13	0
LDO	0	0	0	0	0	1	0	0
Retire	0	0	0	0	0	0	1	0
Resign	0	0	0	0	0	0	0	1

Suppose the inventory for this community on 30 September 2017 was the following:

WO: 700

CWO2: 400

CWO3: 320

CWO4: 250

CWO5: 300

Assume the following r vector: r = [ 1 0 0 0 0 ]

a. As the restricted officer inventory planner, you are charged with attaining the following strength targets for this community for the next four years:

FY18: 2100; FY19: 2200; FY20: 2300; FY21: 2350; FY22: 2400

What should the number of accessions (re-enlistments from the first term population) be each year in order to achieve these targets?

b. Let R = 280 for an indefinite number of years. What is the (entire) long-term or steady-state inventory vector?

c. What is the expected Time In Grade for CWO3?

d. What proportion of CWO3s ever become LDOs?

e. What is probability that a Marine in this community ever becomes a CWO5, given they achieved the rank of CWO2?

f. What is the probability that a particular CWO2 retires in the next year?

g. What is the probability that a member ever retires, given that she attains the rank of CWO3?

h. Consider the transition matrix used in Question above ( Excel Tab ). Suppose you calculated it from last four years of behavior (i.e. 2014, 2015, 2016, and 2017). Describe in words how you would go about validating that the transition probabilities over this time-frame are stationary. What remedial action do you take if you find insufficient evidence to conclude the transition probabilities are stationary?

Answer 1

a)

Here ,

We assume that people who are in LDO, resign and retire is not counted in the strength targets , We can see that in September 2017, the total number of employees in the system is 1970.After one year,we can see that following will be the transition as per the markov's probabilities

Year	2017	2018
WO	700	476
CWO2	400	400
CWO3	320	296.8
CWO4	250	258.2
CWO5	300	311
Sum of employees	1970	1742
LDO	0	59.4
Retire	0	69.4
Resign	0	99.2

This implies that at the end of 2017, 228 people are removed from the strength, So in FY 2018 the number of employees will be 1742 without any new hirings.so hiring/ accession in the year 2018 will be 2100-1742 =358, Assume that the new 358 hirings will be done to the post of WO's, so now we can calculate the new requirement for 2019 similarly for the subsequent years as shown in the table below

Year	2017	2018	2019	2020	2021	2022
WO	700	834	912	976	983	992
CWO2	400	400	427	460	494	518
CWO3	320	297	282	279	284	295
CWO4	250	258	258	254	250	249
CWO5	300	311	322	332	340	345
Strength	1970	2100	2201	2300	2351	2400
LDO	0	59	58	59	61	64
Retire	0	69	71	72	74	75
Resign	0	99	115	125	134	136
Total number of people left		228	244	257	269	275

b) The long term steady state inventory vector will be as shown below , the steady state is acheived when the total people coming into the system become equal to total outflow, Here the total people coming into the system become equal to the people who are retiring, resigning and LDO

WO	875
CWO2	500
CWO3	319
CWO4	277
CWO5	426
LDO	68
Retire	89
Resign	123

c) 6.7 years

We know that if one person join become CWO3 ,probability that he will stay for one year is 0.64, probability that he will stay for 2 years is 0.64*.64

So expected value is given by 0.64+2X 0.64²+...till infinity

which is a infinite AGP. Sum can be calculated using standard equation of infinite AGP. The sum will be 6.7 . This means expected time in grade CWO3 will be 6.7 years

d) .1944

The probability that one person will become LDO from CWO3 is 0.07. if there is one CWO3 , 0.07 will become LDO in first year, In the next year there will be only of CWO3 remaining . Out of which 0.07 will again become LDO , which means 0.07*0.64. So total proportion of CWO3 ever becoming LDO will be

0.07+0.07*.64+0.07*0.64²+.... till infinity which is a GP with a=0.07 r =0.64 .

The sum will be (a/(1-r))=0.1944