A Stochastic Model of Personnel Mobility: Application of Reliability Theory in Manpower System Analysis

A Stochastic Model of Personnel Mobility: Application of Reliability Theory in ManpowerSystem AnalysisAuthor(s): Robert E. StanfordSource: Operations Research, Vol. 33, No. 1 (Jan. - Feb., 1985), pp. 109-125Published by: INFORMSStable URL: http://www.jstor.org/stable/170873 .

Accessed: 08/05/2014 19:19

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

INFORMS is collaborating with JSTOR to digitize, preserve and extend access to Operations Research.

http://www.jstor.org

This content downloaded from 169.229.32.137 on Thu, 8 May 2014 19:19:40 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=informs

http://www.jstor.org/stable/170873?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


A Stochastic Model of Personnel Mobility: Application of Reliability Theory in Manpower

System Analysis

ROBERT E. STANFORD University of Alabama, Birmingham, Alabama

(Received April 1983; accepted December 1983)

Given the distribution of natural losses from any rank of a graded manpower system, and the rate of promotion to higher ranks, we define the mobility of rank members as a function of the cumulative lengths of service of all rank members and those members destined for promotion. We show that when the rank contains many individuals, and promotions are made without regard to seniority, mobility decreases with time in service if the natural loss function has an increasing failure rate, and increases with time in service if the loss function has a decreasing failure rate; mobility increases with time in service if promotions are awarded by seniority. We also derive bounds for the average time in service of rank members, based on the loss function and the promotion rate. In addition, we compare the average time to promotion, average time to resignation, and average time in service for the two promotion policies men- tioned above. Our results show that promotion by seniority produces the longest possible average member service times if the loss function has an increasing failure rate, and the shortest average service times if the loss function has a decreasing failure rate. Using real data, we derive and categorize the theoretical natural loss distribution for a system of university faculty, the implied promotion policy goveming the transition-to-tenure process, and the individual mobility function for nontenured members. This analysis does not require a large amount of data.

IN GRADED manpower systems, the elements of cumulative length of service, rate of departure ("wastage"), and force of promotion are

interrelated factors that determine the flow of rank members through the system and affect member prospects for movement into higher ranks. The total length of time that a new recruit into a rank will remain in the rank is a random variable whose distribution depends on the natural tendency of that member to leave the organization, and on the intensity of promotions from the rank.

The purpose of this paper is to construct a stochastic model of the manpower movement process that relates the distribution of member

Subject classification: 507 mobility in manpower systems, 570 renewal models for manpower service times, 723 failure-rate properties of manpower service times.

Operations Research Vol. 33, No. 1, January-February 1985

0030-364X/85/3301-0109 $01.25 ?3 1985 Operations Research Society of America

109



110 Stanford

service times in the rank to factors and policy decisions that control the flow of persons through the ranks of the system. As a principle objective, we seek to establish a concept of mobility between ranks, and to determine how mobility is affected by promotion policy and the natural loss process. We then derive empirical forms for the characteristics featured in the model using real data collected from a university faculty.

We assume that the loss intensity (from the rank) is a function that is known, that has certain known characteristics, or is a component of the model to be determined from available data. The flow of promotions will be determined by a given fixed promotion rate, which specifies the number of promotions from a rank per unit time per person in the rank, and a promotion rule, which specifies the manner in which a promotee is selected depending upon length of service in the rank. After some important properties of the general model have been established, we shall examine the characteristics of the model for specific promotee selection strategies. We will then use our data to test the assumptions of the model, and to demonstrate how certain characteristics of a manpower system can be obtained indirectly when available data cannot be used to produce the results directly. Specifically, we shall describe the underlying tendencies of the personnel to voluntarily withdraw from the system, the implicit form of the promotion policy, the chances of eventual promotion for an arbitrary candidate, expressed as a function of years (or periods) of accumulated service, and various summary statistics, such as the average lengths of stay in a rank for promoted members and for retiring members.

The structure of our model is based on the renewal model of Bartho- lomew ([1973], Ch. 8), with some modifications. However, in contrast to Bartholomew, we shall not be concerned with inputs and losses to the manpower system as a whole, or the effect of the relative grade sizes on manpower flow, but shall focus on the movement of manpower from one rank to another, and on the factors that affect the mobility of the individual member in this movement process. In another study which utilizes the results of Bartholomew, Robinson [1974] considers a 2-grade special case of Bartholomew's basic model. In some respects, the approach of the analysis in this paper complements my earlier paper (Stanford [1980]), which studied mobility in a discrete-time model with essentially geometric service-time distributions.

1. INDIVIDUAL MEMBER SERVICE TIMES AND MOBILITY

In this section we establish the foundations for developing a model of the flow of manpower from one rank to another. We shall view the flow of persons into and out of a rank as being the result of interactions between two basic random processes acting on each individual member in the rank. The first process-is the natural loss process that determines



Personnel Mobility i11

the random length of stay in the rank by an individual member who is not subject to promotion. The second process is determined by the reoccurrence at random intervals of promotion opportunities for the individual member. Although not essential in'the development of the model, for simplicity we shall assume that all promotions from a rank move only to the next highest rank. Furthermore, we assume that both promotions and natural losses are replaced immediately by new recruits, so that the number of personnel in the rank remains constant. The next section will use the instantaneous replacement of personnel feature. In this section, we are concerned mainly with individual transitions and service times, and we assume that enough time has passed to ensure equilibrium operating conditions within each rank, that is, the random variables that will be used to described individual behavior within each rank are in steady state.

Random variables Z, V and T are defined for each incoming (new) rank member as follows:

Z time that an incoming member will remain in the rank without being promoted (the "natural lifetime" of undisturbed service in the rank),

V time until an incoming member will be' given the opportunity for promotion (dependent on factors external to the rank),

T time that an incoming member will remain in the rank =

min(Z, V). The distributions for Z, V and T are the same for each member and are denoted by F(t), H(t) and G(t), respectively, with F(t) 1 - F(t), and H(t) and G(t) being defined similarly. Furthermore, we assume F has a density f, and that y E(Z) < o. The function F(t) may depend on the promotion policy, which also affects the function H(t), but we assume statistical independence for the random variables Z and V. This condition and the definitions of Z, V and T imply that

G(t) = F(t)H(t). (1) Let Xz(x) f(x)/F(x) denote the failure rate function of Z, and let X, and 0 denote the failure rate functions of V and T, respectively. Then (1) can be expressed equivalently as

OM = XA(t) + MO(t).

If we let a be the probability that an incoming member is eventually promoted, then a = P(Z > V).

Now define the random variables

Tp-time that an incoming member who is eventually promoted will spend in the rank,

TR= time that an incoming member who eventually resigns will spend in the rank.



112 Stanford

By definition of Tp and TR, the distributions Fp and FR of these variables are given by

rt Fp(t) P(T c tIZ > V) = (l/a) fF(x)dH(x) (2)

and

rt FR(t) P(T c tjZ c V) = (1/(1 - a)) H(x)dF(x). (3)

Consequently,

G(t) = aFp(t) + (1 - a)FR(t). (4)

The failure rate functions for Tp and TR are denoted by op and kR. The following definition will formally introduce the concept of rank

mobility. Let

p(x) = Probability that a member who has been in the rank for a time x is eventually promoted

= P(Z>VIT>x).

Note that p(O) = a, and from Bayes' law,

p(x) = aFp(x)/G(x). (5)

The function p(x) will be referred to as the mobility function. We shall first present some general properties of p(x) for any manpower system; in the next section we examine this function under various assumptions regarding the loss function for rank members and the promotion policy applied to the rank.

The mobility function p(x) supplies the link between the failure rate functions previously defined, by means of the relationships

op(x) = '(X)/p(X), OR(X) = Xz(x)/(1 - p(x)). (6)

We have the analog

+(X) = p(X)qp(X) + (1 -P(X))qR(X)

of Equation 4. Assumming p(x) is differentiable, we obtain

p'(X) = p(X)(l - P(X))(OR(X) - MO,

which implies that p(x) is increasing in x in some neighborhood of any point where the failure rate of TR is greater than the failure rate of Tp. In the next section we study special cases where these failure rates can be compared directly. The cumulative length of service distributions are related by

FP(X) = FR(x)p(x)(1 - a)/(a(l - p(X))). (7)



Personnel Mobility 113

From the equivalence of a distribution and its failure rate function, and the relationships (5), (6) and (7), we have the following result which establishes equivalent means of describing the individual member characteristics considered thus far.

PROPOSITION 1. The functions p(x) and F(x) determine FR(x), FP(x) and a, and vice versa.

Besides demonstrating the impact of rank mobility in determination of the time-in-service distributions of both promotees and resigners, Proposition 1 establishes that the loss function F(x), when not known, can be determined from data describing the distributions FR(x) and FP(x) and the value a.

2. PERSONNEL TURNOVER AS A RENEWAL PROCESS

The previous section considered individual member service times and opportunities for promotions. Promotion, however, is a process defined on the rank as an entire body, from which some members are selected over others because of characteristics that distinguish them from the entire group of persons in the rank. This section will develop the relationship between individual member characteristics and rank characteristics so that effects of rank promotion policy on the individual mobility can be determined.

The number of persons in the rank, denoted by N, will remain constant over time. We regard the rank as consisting of N "slots" which are being vacated periodically by either member promotion or resignation, and immediately filled by new recruits. Furthermore, we adopt Bartholomew's (p. 210) assumption that the movement of members into and out of each slot is independent of movements in all other slots, so that the completed lengths of member service constitute a renewal process. The flow of persons out of the system is then determined by the pooled output of N identical renewal processes.

Now the replacement of persons promoted from a particular slot also constitutes a renewal process. Let hp(x) represent the renewal density for the promotee replacements and h(x) the renewal density for all replacements, and let

P lim,O hp (t), h lim,O h(t),

both of which exist from the basic properties of the renewal processes they represent.

Note that the time between replacements in a slot is equal to T, the total time in service of an arbitrary member. Hence ift E(7), we have

h= l/M (8)

The following analysis will formalize the relationship between the



114 Stanford

renewal processes defined above and the individual characteristics devel- oped in the previous section. Let

Tp --time between promotions from a slot M number of members that will occupy a slot before a promotion

occurs from the slot.

We let the random variables Zi, Vi and TR', which form independent sequences and are distributed as Z, V and TR, respectively, correspond to the ith customer to occupy a slot following a promotion. The distribution of Tp is denoted by Fs(t). Now

P(M = n) = P(Z1 c Vl, Z2 C V2, Zn > Vn) (9)

- a(1 - a)n1.

Furthermore,

P(rP C t I M = n) = P(TR1 + TR2 + n+T + TP < t). (10)

Combining (9) and (10) and taking Laplace transforms gives a relationship between the promotion renewal process and the CLS distributions for promotees and resigners:

Fs(s) = aFp(s)/(1 - (1 - a)FR(s)). (11)

Taking derivatives in (11) and using the relationship 1/P = E(TP) =

-Ps'(0) and utilizing (8), we obtain the result

a = 1P. (12)

The quantity P is referred to as the promotion rate; by our definition, each job slot is under a force of P, so that the rate of promotions from the entire rank is NP. In our interpretations, P is determined a priori, as the mean value of a steady-state process arising from the vacancies occurring in higher ranks, or by the desire to provide a certain degree of promotion opportunities for rank members. In any case, P will be regarded as a parameter that interacts with other policy factors concerning promotion to determine the flow of promoted personnel out of the rank.

Note that in view of relations (12) and (4), we may replace a by P in Proposition 1.

The mean residual life of the distribution F is defined as

q(t)= F(x)dx/F(t); (13)

F is said to be decreasing mean residual life (DMRL) if q(t) is a decreasing function of t, and increasing mean residual life (IMRL) if q(t) is increasing in t. Note that q(t) = 1/Xe(t), where Xe(t) is the failure rate function for Fe, the equilibrium distribution of F.




Now it is easy to show that

e' (t) = Xe(t)(Xe(t) -X M(t)). (14)

Furthermore, if F is IFR (DFR), we have 00 00 r -1

Xe(t) = (I' (F(x)/F(t))dx) -(') (J f (/(x)/f(t))dx) = (t),

hence Fe is IFR (DFR), and therefore (14) implies that F is DMRL (IMRL).

Recall that y = E(Z). The following result holds for any promotion policy.

PROPOSITION 2. If the distribution F is DMRL (IMRL), then

A ?-(C) y/(1 + Py). (15)

Proof. ,u = fS F(x)H(x)dx = y 'fO fe(x)H(x)dx = y f' Fe(x)dH(x). From (14), if q(t) is decreasing (increasing), then Se(t)- () Xz(t), which implies that Fe(t) ?-(c) F(t), hence ,u-(c) y(l - a). From (12), a = AP, which implies (15).

3. EFFECTS OF EXPERIENCE ON PROSPECTS FOR PROMOTION

Another policy factor that will affect the length of stay of members in a particular rank is the consideration given to accumulated service time in that rank, when the selection of a promotee is made. In this section, we derive explicit forms for the mobility function and the various service time distributions when specific promotion policies are applied to the rank, and identify the important characteristics of these forms.

We assume that the rate of promotion P is determined previously by a pattern, perhaps random, of promotions that are themselves determined by factors external to the particular rank under study.

Organizations may emphasize rank experience differently in the promotion process. Following Bartholomew and others, we will consider here the two extreme cases of (i) promotion awarded to the most experienced rank member, and (ii) promotion awarded at random with respect to experience in the rank. Strictly speaking, our results apply for ranks with a large number of members that have reached equilibrium operating conditions. Bartholomew has pointed out that these assumptions apply well for finite ranks numbering at least 30 members, but that equilibrium distributions for the member service times tend to be approached rather slowly.

We seek expressions for p(x), which depends on the function H(t). The distribution H(t) for both promotion policies to be considered can be determined directly by utilizing processes, similar to those presented and employed by Bartholomew in Chapter 8, Sections 2 and 3, which



116 Stanford

deal with length of service in the overall system; however, we shall apply the more general results from Sections 4 and 5 of the same chapter, which deal with rank service times, and deduce H(t) indirectly.

For promotion at random with respect to length of service in the rank, from relation (8.79) of Bartholomew we obtain the expression

G(t) = F(t)e-pt,

hence H(t) = ePt, and

rt rt Fp(t) = (1/a) f F(x)Pe-Pxdx = (1/u) G(x)dx. (16)

Also, since FA(t) = Ge(t), the equilibrium distribution of G, relation (5) becomes

00 p(t) = P/kp(t) = P f F(s + t)e-Psds/F(t). (17)

PROPOSITION 3. In large ranks where promotions are made at random with respect to length of service in the rank, p(t) is an increasing (decreasing) function of t if F is DFR (IFR).

Proof. From (17), P(t) increases in t if op(t) decreases in t; (16) implies that Fp is the equilibrium distribution of G, hence if G is IFR (DFR), then Fp is IFR (DFR). Since 0(t) = X,(t) + P in this case, the result follows.

For promotion of the most experienced member, expression (8.69) in Bartholomew, in effect, states that

G_t)= JF(t) if t < L

lM to if t > L,

where L, the maximum attainable length of service in the rank, is determined by

rL

F(L) = P P(x)dx, (18)

which is analogous to relation (8.63) in Bartholomew. Then

JI if t<L Fp (t) =- HI(t) =I

o iif t-L,

and p(t) a/F(t) = F(L)/F(t) for t < L. (19)




The following result, which is analogous to Proposition 3, is an im- mediate consequence of relation (19).

PROPOSITION 4. In large ranks where promotion is given to the most experienced rank member, p(t) is an increasing function of t.

4. EFFECTS OF PROMOTION POLICY ON AVERAGE SERVICE TIMES

We shall now compare some of the operating characteristics of the manpower system under the two promotion strategies considered previously.

Let

A age in the rank (i.e., experience accumulated in the particular rank) of an arbitrary member,

00

and v-E(A) = (1/,u) xF(x)H(x)dx.

The promotion prospects of members already in the system are measured, in an average sense, by

00 r-E(p(A)) = (a/lt) f Fp(t)dt = P(E(Tp)). (20)

Relation (20) is analogous to the expression (12), a = ,AP, which measures the promotion prospects of incoming members. Both (12) and (20) show that, for a fixed promotion rate P, promotion policies that increase longevity in the rank will increase the overall mobility of rank members. Hence, greater prospects for promotion are not incompatible with greater average levels of experience. Of course, if P increases, it is intuitively clear that for any promotion strategy both ,u and v will decrease, since promotion of any member is followed by instantaneous replacement by a new recruit with no experience. At the same time, a increases as P increases. These effects are easily verified analytically.

For any promotion policy (3) gives

FR(t) = (1/(1 - a))P(V > Z > t) c (1/(1 - a))P(V - Z)P(Z > t),

which implies

PR -E(TR) <y.

Computation of expected values from (4), in combination with (12), gives

tip E(TP) < 1/P



118 Stanford

for any promotion policy. Propositions 3 and 4 can be applied to relation (7) with the results

YP 1 Y . PR (21)

for promotion by seniority and for random promotion when F is DFR, and

YP C 9 C PR (22)

for random promotion when F is IFR. In view of the established links between rank tenure and mobility it is

desirable to compare the average times in service of rank members for various different promotion rules.

For a given loss distribution F(x) and promotion rate P, we shall be concerned with finding the maximum of the expression

P= J'F(x)H(x)dx

over all survival functions H(x) = P( V > x) satisfying relation (12), expressed in the form

1- f H(x)f(x)dx = Jo F(x)H(x)dx P. (23)

PROPOSITION 5. In large ranks promotion by seniority produces the longest possible overall member service times if F is IFR, and the shortest possible overall service times if F is DFR.

Proof. Let D(x) = 1 for x < L and D(x) = 0 for x >- L; in other words, 1 - D(x) is the promotion-by-seniority distribution function for V. Note D(x) satisfies (23) by definition of L (relation (18)). Define

R(x) F(x)/(F(x) + f(x)/P) = P/(P + kz(x)), 00

and F(X)(D(X) - H(x))dx.

Note R(x) is increasing (decreasing) if F is IFR (DFR). From (23), we have

(F(x) + f(x)/P)D(x)dx= 1/P 0 (24)

= f,> (F(x) + f (x)/P)H(x)dx.




Using the definition of D(x),the monotonicity of R(x), and (24), we have

rL

A= fL R(x){F(x) + f (x)/P}(D(x) - H(x))dx

+ f R(x){F(x) + f (x)/PI(D(x) - H(x))dx L

-(c) R(L) LI {F(x) + f (x)/P}(D(x) -IH(x))dx

+ J {F(x) + f (x)/P} (D(x) - H(x))dx = R(L) = O 0

if F is IFR (DFR).

If , denotes E(T) when promotion is by seniority, with ,u representing arbitrary average time-in-service, Proposition 5 states that F IFR (DFR) implies that ,i-(c) ,u. Then, applying (21) and (22), we find that if F is IFR, the time to promotion is longer under promotion by seniority than under promotion at random, that is

Yp ' / ' y liP,

where Aip corresponds to promotion by seniority and ,i and jip correspond to promotion at random. Also note 'p = L, and jip = expected age in rank under promotion at random.

Let 0' expected age in rank under promotion by seniority. We have the following proposition which is analogous to Proposition 5.

PROPOSITION 6. In large ranks, v>- when F is DFR and when F is IFR and the function Q(x) xP/(P + X,(x)) is increasing in x.

Proof. (i) Using the definition of D(x) gives f' (x - L)F(x)D(x)dx c f (x - L)F(x)H(x)dx, hence, from the definitions of v and pi, it follows that

(V/ - L) c (p - L). (25)

Since F DRF =, ,u , and since io c L, the result follows when F is DFR.

(ii) Following the method of proof in Proposition 5, we write

AP - =f Q(x)(F(x) + f (x)/P)(D(x) - H(x))dx, (26)

hence if Q(x) is increasing in x, the right-hand-side of (26) is cO. Since F IFR implies that , A ,t, the proof is complete.



120 Stanford

Proposition 6 indicates that under certain conditions, with F IFR, it is possible to have promotion by seniority produce longer average total times-in-service (i.e., p) and, at the same time, produce lower average levels of experience in the rank (i.e., v). This situation will occur, for example, if F(x) is the Weibull distribution with parameters 1 < 0 < 2, X > 0, in its failure rate function X,(x) = X0xx1.

For an example with F IFR and i < v, consider the degenerate distribution with F(x) = 1 for x < u, F(x) = 0 for x > , where u satisfies 1/P < a < ln 3/P. In this case, io = L/2 = 1/(2P). A comparison with promotion at random, where Hi(x) = ePx, yields

ii = 1/P - o7e Pa//(1 - e-) < P.

5. ANALYSIS OF DATA FROM A UNIVERSITY FACULTY

Our data describing the process of awarding tenure at a relatively small academic institution were collected from administrative files that con- tained records of practically all personnel movements since the founding of the institution. We sought to study the promotion-to-tenure process during a period in the 1970s when the faculty size remained more or less constant and the distribution of faculty among the ranks in the system was stable.

Career records of members who left the institution for any reason from the nontenure ranks, and records of members who were granted tenure, were examined for three consecutive years during which the total head- counts of nontenured faculty were 121, 117, and 107, respectively, and there were 10, 10, and 11 promotions to tenure, respectively. Data were then aggregated and the numbers of years served in nontenure tabulated for both groups of faculty. Although times-in-service were recorded for each individual faculty member promoted or resigning from nontenure over the 3-year period, this period represented only a cross section of the system's activity over time. However, by applying results from the model (and regarding the underlying assumptions of the model as valid), it was possible to derive forms for cohort characteristics such as a and p(x), which could not have been determined directly without examining much more data.

Our time-in-service data included 3-year aggregated values for nip no. of promotees who spent i years in nontenure and niR no. of nonpromotees who spent i years in nontenure, where 1 c i c 8. (Eight years is the standard maximum term that a faculty member may remain employed in a nontenure capacity, according to the American Association of University Professors).

According to our model, the eventual promotion of members entering nontenure is mathematically determined a priori by events of the type IZi > Vi}. Now, the members who were "predestined" for promotion are




not actually revealed until these members leave the system as promotees. Therefore, if the size of the nontenure rank remains fairly constant, so that the number of faculty entering nontenure each year approximately equals the number leaving, we can produce reasonable estimates for our time-in-service distributions using the relations

P(Tp = i) = nwp/np, P(TR = i) = niR/nR,

and P(T = i) = (nip + niR)/(np + nR) (27)

where np = nip and nR = i=j njR. Using the relations (27), we obtained ,Ap = 5.74, LR = 4.06 and it = 4.70 for our average times in service. Table I includes the values of Fp(X), PR(X) and G(x), and Figure 1 is a graphical comparison of these values.

1O F(x) FRx)

A

A

x 1 2 3 4 5 6 7 8

Figure 1. Fitted curve graphs of observed time-in-service distributions.

In the discrete case, which applies for our faculty data, the quantities P and h from Section II are given by P = lim,O (no. promoted in period t)/N and h = limt,. (no. promoted and resigning in period t)/N, where N is the (fixed) nontenure rank size. Using (8) and (12), it follows that

J no. promoted in period t -

tno. promoted and resigning in period tf

hence an appropriate estimate for a is

a = np/(np + nR). (28)

Our data yielded a value of 0.383 for a'. If P is computed from 'a and A via relation (11), we obtain a value of

0.0815. The direct, aggregated data-computed value of P is (10 + 10 + 11)/(121 + 117 + 107) = 0.0899, which differs by about 10% from the first figure for P. This difference is apparently a result of the lack of



122 Stanford

TABLE I

OBSERVED DATA AND DERIVED FORMS

x FP(X) FR(X) C(X) F(x) H(x) XA(x) XA(x) p(x)

0 1.000 1.000 1.000 1.000 1.000 - - 0.383 1 1.000 0.860 0.914 0.914 1.000 0 0.086 0.419 2 1.000 0.740 0.840 0.840 1.000 0 0.081 0.456 3 0.968 0.580 0.728 0.741 0.983 0.017 0.118 0.509 4 0.807 0.360 0.531 0.603 0.881 0.104 0.186 0.582 5 0.484 0.240 0.333 0.519 0.643 0.270 0.139 0.557 6 0.355 0.180 0.247 0.461 0.536 0.166 0.112 0.550 7 0.129 0.100 0.111 0.369 0.301 0.438 0.200 0.445 8 0 0 0 0.164 0 1.000 0.556 -

stability in the rank size during, and possibly prior to, the period of time associated with our data.

The discrete versions of relations (2) and (3) yield the formulas

P(V = k) = aP(Tp = k)/I?=k+1 P(Z = i) (29)

and P(Z = k) = (1 - a) P(TR = k)/>IZLk P(V = - (30)

which define a recursive procedure for computing the distributions F and H, beginning with the result P(Z = 1) = (1 - a)P(TR = 1). The results are given in Table I, and graphed in Figures 4 and 5.

Using the estimates defined in (27) and (28), relation (4) is verified identically. With F and H computed via relations (29) and (30), using the estimate for a in (28), Equation 1 is also an identity relationship.

The mobility function p(x) for new appointees to nontenure is shown in Figure 2. The shape of the curve indicates that a nontenure faculty member's chances for eventual promotion (considered from the perspec- tive of the overall university) increase with experience up to midcareer evaluation, then decline as the maximum allowed term of 8 years in

p (x)

56 .5 .4

1 2 3 4 5 6 7 Figure 2. The mobility function for individuals.




nontenure is approached. This feature will be discussed further when the form of the promotion policy is examined.

Computed values for the discrete-time failure rate functions X,(k) = P(V = k)/Ek P(V = 1) and X,(k) = P(Z = k)/>2=k P(Z = i) are given in Table I and graphed in Figure 3. Both X, and X, appear fairly constant during the earlier years in nontenure, but fluctuate and intersect around year 5. To explain this phenomenon, we should recall our basic assumption of independence between the random variables Z and V. In reality, it might be expected that individual faculty perceptions concerning the pattern of arising opportunities for promotion, i.e., the form of H(x), would affect the tendency of these faculty members to leave the system. In such situations, an increasing failure rate for H would be accompanied by a decreasing failure rate for F.

.5

.4 XV(x)) XZ(x)

.3

.2

z /

~~~~~~~~~~x 1 2 3 4 5 6 7 8

Figure 3. Failure rate functions.

Computation, not presented here, of q(t) using relation (13), indicates that F is DMRL. Now -y = E(Z) = 5.609, and Proposition 2 readily verified for our data.

We can determine to some degree the emphasis that has been placed on seniority as a qualification for promotion by comparing our empirically derived distribution H to the distributions that would result if our derived form for F and value of P were applied to a promotion process determined by seniority only, denoted by H, and without regard to seniority, denoted by H. Then 1 - H(x) = e-0.09x and 1 - H(x) = 1 if t < L, and equals zero otherwise, where L satisfies (17) with P = 0.09 and the form of F(x) given in Figure 4. In this case, L 6.07, and our comparison is shown in Figure 5. The relative positions of the curves make it possible to express H(x) as an approximate convex combination of 1 - H(x) and 1 - H(x), implying that experience enhances promotion prospects, but does not inevitably lead to promotion.



124 Stanford

F (x)

1.04

x 1 2 3 4 5 6 7 8

Figure 4. Fitted-curve graph of the derived underlying loss distribution.

1.0 *- x)

\_ +\H(x) |~ IH(x v 1 - H(x

x 1 2 3 4 S 6 78

Figure 5. Comparison of the implied actual promotion opportunity pattern with extreme policy patterns.

6. DISCUSSION

In the previous section, we applied the results of our model to describe the interaction of factors affecting the promotion and loss processes in the nontenured portion of the particular university faculty. Using the structure of our model, we can also forecast some of the effects of changes in promotion policy and shifts in the underlying loss distribution, and gain greater insights regarding future patterns of behavior for our system than could be obtained by simply extrapolating from historical data.

Proposition 1 implies that if a specific form of F has been determined (or assumed), various characteristics of the overall manpower system can be derived assuming alternative forms for the individual mobility function p(x). This approach may make it possible to examine the relationship between the promotion opportunities afforded individual members of the system and the objectives of the overall manpower system, and hopefully




to develop promotion policies that simultaneously meet the needs of the system and the individual.

It is possible to generalize our device of regarding the time T spent in rank service as the minimum of two other random variables, Z and V, by supposing that

T = min(X1, X2, *., Xn), (31)

with Xi - Fi for i = 1, 2, ... , n. Some of the Xi would correspond to alternative processes causing members of the rank to leave, and the remaining Xi would correspond to alternative types of promotion opportunities for the rank members. In this case, assuming pair-wise independence of the Xi, we have G(t) = HII=L Fi(t). Furthermore, relation (4) of Section 1 generalizes to

G(t) = En l ajP(T c tIXi > Xj for all j $ i), (32)

where ai P(Xi > Xj for all j $ i). As a final point for discussion, we consider the fact that the mixed-

exponential distribution, a convex combination of distinct exponential distributions, has frequently been found (see Bartholomew, Chapters 6, 7 and 8) to describe the cumulative lengths of service (CLS) of members of particular manpower organizations. Perhaps there is an explanation for this phenomenon in the structure of our mobility model. If the CLS of members in an organization, or in a rank of an organization, is indeed determined by several competing factors, as described by relation (31), these effects can be captured in the form of the CLS distribution function in the manner of relation (32), for which the mixed-exponential distribution would qualify as a special case.

REFERENCES

BARTHOLOMEW, D. J. 1973. Stochastic Models for Social Processes, Ed. 2. John Wiley & Sons, London.

ROBINSON, D. 1974. Two-Stage Replacement Strategies and Their Application to Manpower Planning. Mgmt. Sci. 21, 199-208.

STANFORD, R. E. 1980. The Effects of Promotion by Seniority in Growth- Constrained Organizations. Mgmt. Sci. 26, 680-693.



Documents

A Stochastic Model of Personnel Mobility: Application of Reliability Theory in Manpower System Analysis