Asymptotic behavior of the stock vector in a mixed push-pull manpower model

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1 Asymptotc behavor of the stoc vector n a mxed push-pull manpower model Mare-Anne Guerry, Tm De Feyter Abstract In ths paper, the asymptotc behavor of the tme-homogeneous mxed push-pull manpower model s studed under the assumpton that the desred stoc vector and the recrutment polcy are fxed over tme. In the mxed push-pull manpower model, the nternal moblty of a personnel system can be regulated by both pull and push transtons. Based on those characterstcs, we express and examne the dynamcs of the personnel system by formulatng the mxed push-pull manpower model by means of partcular transton matrces, whch we demonstrate to have nterestng propertes. We show that under certan condtons the stoc vector converges. An explct analytcal form for ths lmtng personnel stoc vector s found. Keywords: Manpower plannng; stochastc models; push models; pull models; asymptotc behavor.. Introducton Manpower plannng provdes long term organzatonal decson support regardng recrutments, redundances and nternal staff moblty. Based on aggregate analyses, varous mathematcal models for manpower plannng have been developed to descrbe, predct and control the future personnel avalablty n the dfferent states of a manpower system [,2]. The total populaton s therefore classfed nto homogeneous groups of employees, accordng to varous employee characterstcs, le grade, slls, nowledge and abltes [3-5]. The evoluton of personnel avalablty fully depends on the recrutment and wastage n each group and on the personnel moblty among those groups. Hence, to study ths personnel dynamcs, organzatons are modeled as systems of stocs and flows. Although recently alternatve approaches have been used (e.g. [,6-8], most mathematcal models for manpower plannng are founded on Marov theory [9-]. Those so-called push models assume that all employees wthn a homogenous group are subject to the same nternal

2 transton and wastage probabltes. In general, three types of push models can be dstngushed. Frst, tme-homogeneous Marov models are sutable for organzatons n whch the personnel flows among groups can be defned as tme ndependent transton probabltes (e.g. [2,3]. It s well nown that the flows follow a multnomal dstrbuton, for whch the parameters,.e. the transton probabltes, need to be estmated [4]. Second, non homogeneous Marov models relax ths assumpton and nclude tme dependent transton probabltes as model parameters (e.g. [5-8]. Fnally, sem-marov models relax the assumpton of homogeneous subgroups, by consderng condtonal transton probabltes, dependng on the ndvdual s duraton of group membershp (e.g. [9-2]. Once hred, employees grow n terms of slls, nowledge and abltes. Hence, tme-dscrete push models are very sutable to model personnel flows, snce n each tme nterval a certan number of employees s expected to mae a transton among states determned by these personnel characterstcs. Whle consderable research has been devoted to push models n manpower plannng, less attenton has been pad to the so-called pull models. Based on Renewal theory, n pull models, a transton nto a state s only possble f there are vacances to be flled [2]. The number of vacances s thereby assumed to follow a bnomal dstrbuton, based on the wastage probablty n the recevng state. In practce, a mx of push and pull transtons mght occur n the same personnel system. Therefore, Georgou and Tsantas [22] ntroduced a hybrd model, whch was further developed by Dmtrou and Tsantas [23,24]. Ths approach allows modelng push as well as pull flows wthn the same personnel system. Besdes push flows between actve classes, a preparaton class s ntroduced from whch ndvduals can be pulled towards the actve classes n case vacances arse n the actve states. However, ths approach s restrcted to an embedded Marov model, assumng nown total system sze, mplyng that the desred total number of ndvduals n the system s nown and the vacances are determned at an aggregated level. In certan organzatons however, the desred number of employees and the number of vacances are specfed at group level. De Feyter [25] therefore ntroduced the mxed push-pull manpower model, whch allows modelng push as well as pull flows between all states n the personnel system, by consderng vacances n the ndvdual states. Accordngly, the mxed push-pull 2

3 manpower model s a generalzaton of the tradtonal push and pull models. The model s very sutable for organzatons n whch the states are defned among others by the grades n the personnel herarchy. Ths way, the mxed push-pull approach allows modelng personnel transtons that tae place because of vacances arse at hgher levels n the personnel system (pull flows or employees attan slls, nowledge and abltes that are necessary for grades at a hgher level n the herarchcal system (push flows. Hence, the flows can be determned by push and pull mechansms. Consequently, n comparson to tradtonal push and pull models, n a mxed push-pull model, t s harder to examne the future personnel dynamcs. For a state n the personnel system, dependng on the vacances n other states n the system, the mechansm (push, pull or push-pull that determnes the flows may dffer over tme. In some tme ntervals, when there are vacances at hgher levels, both push and pull promotons would occur, whle n other tme ntervals, when there are no open postons to be flled n hgher levels, people would only be promoted by the push mechansm. Furthermore, the ntervenng transton mechansm can be dfferent for the dstnct states of the manpower system. A central ssue n manpower plannng research s the attanablty and mantanablty problem [23,26-30]. For a manpower system, one of the objectves s to control the evoluton of the personnel system n order to attan or mantan a desrable stoc n the future. Another mportant concern n manpower plannng s the drecton n whch the personnel structure s changng [24,3]. Therefore, research n manpower plannng s very nterested n the asymptotc behavor of personnel systems. For the mxed push-pull manpower model, De Feyter [25] dscussed the mantanablty and attanablty problem under control by recrutment. Furthermore, some prelmnary results were found on the asymptotc behavor of the tme-homogeneous mxed push-pull model, under the assumpton that the worforce demand and the recrutment polcy are fxed over tme. For organzatons n whch the transton mechansm s determned by both pull and push flows n all states of the personnel system and durng all tme ntervals, t s shown that under certan condtons the stoc vector converges. The numercal llustratons however reveal that these are suffcent but not necessary condtons for convergence towards a lmtng stoc vector n mxed push-pull personnel systems. 3

4 In the present paper, we further elaborate on the asymptotc behavor of the mxed push-pull manpower model. Frst, we extend the nvestgaton by relaxng the restrcton that for each state push as well as pull flows are nvolved. Second, we allow the ntervenng transton mechansm to dffer between the states of the manpower system. Ths paper proceeds wth a noton of the mxed push-pull model n the next secton. A set of dfference equatons s presented, as well as ts specfc notatons and propertes, what allows descrbng the evoluton of the stocs n a mxed push-pull approach. In Secton 3, by ntroducng specfc transton matrces M ( υ (, we propose a reformulaton of the mxed push-pull model. As s shown n Secton 4, these transton matrces have some nterestng propertes, whch are very useful to examne the asymptotc behavor of the mxed push-pull model n Secton 5. Fnally, Secton 6 provdes an llustraton of the theoretcal results of the precedng sectons. 2. The mxed push-pull model Ths paragraph provdes the mxed push-pull as presented n De Feyter [25]. For a personnel system dvded nto homogeneous groups, the stocs at tme vector n( ( n (,..., n ( t are denoted by the row = wth n ( beng the expected number of employees n homogeneous group at tme t. Let W ( be a dagonal matrx wth W ( = w ( the wastage probablty n group n tme nterval [ t,t. Further, we defne the row vector V ( ( V (,..., V ( = wth V ( beng the number of vacances n group to be flled n tme nterval [ t,t. The mxed push-pull model assumes the vacances to be V ( = Max{ 0, n ( n( t. [ I W ( ]} wth ( n ( n ( = beng the row vector wth n ( t the desred number of employees n group at tme t. The Max-operator s hereby defned as Max { A, B} = ( Max{ A, B},..., Max{ A, B }. The nternal dynamcs of the personnel system n the mxed push-pull model s frstly regulated by pull transtons to fll the vacances, governed by the matrx S( = ( s ( wth ( j s j 4

5 beng the probablty n tme nterval [ t,t from group j. for a vacancy n group to be flled by an employee Remar that for every group j and at each tme t the stoc should be suffcently large to capture all pull transtons from that group,.e. [ w j ( ] j, t : n ( n ( t. V (. s ( ( j j j Note that S ( s not necessarly a row stochastc matrx. Indeed, the mxed push-pull model assumes that not all vacances need to be flled by nternal transtons, but can also be captured by pull recrutments. Subsequently, the nternal moblty of the mxed push-pull model s supplemented by push transtons. The push flows are characterzed by the transton matrx = ( p (, and by the row stochastc transton matrx Q( ( q ( P( t j =. Its elements ( represent the transton probabltes for employees n group at tme t to be n group j at tme t, under the condton that they do not leave the system nor mae a pull transton durng tme nterval [ t, : [ I W ( ]. Q( = P( (2 j q j The total number of recrutments addtonal to the pull recrutments at tme t s R ( and r ( s the proporton of R ( assgned to group. The vector. row vector r( ( r ( = s the recrutment The expected number of personnel n every homogeneous group can be calculated by a system of dfference equatons: { n( t [ I W ( ] V ( S( } Q( R( r( { 0, n ( n( t [ I W ( ]} n ( = V ( t (3 V ( = Max (4 w ( = p ( for =,..., (5 j 5

6 3. A reformulaton of the mxed push-pull model Accordng to (4, there are vacances n group to be flled n case V ( > 0 n ( n ( t. [ w ( ] > 0 By ntroducng for all {,..., } the matrx I satsfyng ( I = for ( j, l = (, and ( I = 0 for ( j, l (, jl and by settng { V ( 0 } υ ( = > t A( = I S(. Q( M( υ ( = [ I W( ] Q( I A( t υ ( (3 and (4 can be wrtten n terms of the matrces I, A( and ( υ ( : jl M Theorem. Proof. V ( = [ n ( n( t ( I W ( ] I υ ( n( = n( t. M ( υ ( n ( I. A( R( r( (7 υ ( On the one hand, for υ (t holds that V ( > 0, what s equvalent wth n ( n ( t.( w > 0 [ n ( n( t ( I W ( ]. Moreover, by defnton of the matrces I, the -th coordnate of I υ ( t [ ] = n ( n ( t.( w equals n ( n( t ( I W (. On the other hand, for υ (t holds that the -th coordnate of Max{ 0, n ( n( t. ( I W ( } as well as of n ( n( t ( I W ( equals zero. Consequently Max Whch proves (6. [ ] { 0, n ( n( t. ( I W ( } = [ n ( n( t ( I W ( ] I υ ( t I υ ( t (6 6

7 Accordng to (3 the evoluton of the stoc vector can be expressed as: n ( = V ( { n( t [ I W ( ] V ( S( } Q( R( r( = V (. [ I S( Q( ] n( t [ I W ( ] Q( R( r( By (6 and by the defnton of the matrces A ( and ( υ ( ths expresson can be rewrtten as: n ( = n ( n( t [ I W ( ]} M { I.[ I S( Q( ] n( t [ I W ( ] Q( R( r( υ ( = n( t [ I W ( ] Q( I.[ I S( Q( ] n ( I. [ I S( Q( ] R( r( υ ( t υ ( = n( t [ I W ( ] Q( I. A( n ( I. A( R( r( υ ( υ ( = n( t. M ( υ ( n ( I. A( R( r( Whch proves (7 and the Theorem. υ ( The convergence propertes of the stoc vector n( wll be dscussed n terms of the matrces ( υ (. Therefore n frst nstance these matrces are examned n the next paragraph. M 4. Propertes of the matrces ( υ ( M υ = and A( = I S(. Q(, t holds that: Snce M( [ ] ( I W( Q( I A( υ ( M( υ ( = [ I W( ] I I S(. Q( I (8 υ ( υ ( The matrces ( υ ( are composed of the transton probabltes characterzng the push M model, n terms of W ( and Q (, and of the pull transton matrx S (. For ths reason we wll call the matrces ( υ ( the mxed push-pull (MPP matrces. Notce that for υ (t the -th M 7

8 row of ( υ ( equals the -th row of the push transton matrx P. In the partcular case M υ ( = { } holds that M ( υ ( = P(. In the stuaton of homogenous personnel groups, the number of MPP-matrces ( υ ( υ equals =!( ( ( {,..., }!. 0! M ( satsfes the followng propertes: υ υ An MPP-matrx M = I W I IS. Q I = ( mj Property. The non-dagonal elements are between 0 and 2: 0 m 2 j j Property let us conclude that an MPP-matrx satsfes the condtons to be an ML-matrx [32]. However an MPP-matrx has some addtonal characterstcs: Property 2. The dagonal elements are between - and : m Property 3. All the row sums are between 0 and : 0 m j Proofs. M = j = ( I W Q I.( S. Q I and denotng Kronecer delta by δ j, υ υ Snce ( m : m = w [ q ( S. Q δ ] for υ for { } j ( j j j and mj = ( w.,..., \ υ : m j = ( w. qj and mj = w ( S. Q These expressons prove the propertes, 2 and 3, as Q s a row-stochastc matrx and the pull transton probabltes satsfy 0 s j and 0 s j, j. j 8

9 In Bartholomew et al. [2] the lmtng behavor of the push model s studed under the assumpton that p <,.e. the wastage probablty w s dfferent from zero. Because t s j mpossble that wastage never occurs, manpower systems are open systems. Hence, ths assumpton s reasonable. However, t s not for all personnel groups a necessary condton for examnng the convergence of the mxed push-pull manpower system. For the further analyss of the MPP-matrx M n ths paper, for each personnel group n whch vacances arse, t s suffcent to assume that ether the personnel strategy does not prescrbe that all vacances are entrely flled by pull promotons or that the wastage probablty n that group s dfferent from zero. Assumpton. I υ : 0 w {,..., } \ I s < I j Property 4. If Assumpton s satsfed, then for an MPP-matrx M holds: : 0 m j <. Proof. The row sums of the matrx M satsfy: and m m j j = ( w. ( S. Q = ( w. s. q = ( w. s for υ = w j,..., for { } \ υ l= Ths let us conclude under assumpton that m <. Whch proves property 4. j l lj l= l 9

10 Lemma. If Assumpton s satsfed, the nverse matrx ( I M exsts. Proof. The exstence of the nverse matrx ( I M wll be proved by showng the ndependency of the rows of the matrx frst row of I M. Wthout losng generalty we wll prove that the I M can not be wrtten as a lnear combnaton of the other rows of ths matrx: A soluton ( C 2 C3... C of the followng system of lnear equatons E E2... E m m2 M m = = = 2 C C 2 C ( m 2 m 22 M m M... C C m m M C ( m 2 satsfes for each subset I { 2,...,} : m = j C mj C as a result of addng up all the equatons { 2,..., } I j I m (9 j I j I I j I E j wth j I. Hereby the notaton I refers to the set I = \. Snce 0 m j < (Property 4 and the non-dagonal elements m j 0 ( j are non-negatve (Property, for all j I I, the coeffcent of C s strctly negatve: m < 0. For all I = { 2,..., n} \ I the coeffcent of C s postve: m 0. The fact that j the left member of equaton (9, j I m j, s postve, results n the concluson that for a soluton C C... C of the system t s not possble thatc > 0 for all I and C 0 for all I. ( 2 3 Snce ths reasonng holds for all subsets of ndces I { 2,...,} system has an empty soluton set. For ths reason the frst row of combnaton of the other rows of ths matrx. More generally, the rows of ndependent. For ths reason ( I M exsts. Whch proves the Lemma. j I j, there can be concluded that the I M s not a lnear I M are lnear I M s a non-sngular matrx for whch the nverse matrx 0

11 5. Asymptotc behavor of the mxed push-pull model We nvestgate the asymptotc behavor of the mxed push-pull model under the assumpton of tme-homogenety and a stablzed stoc evoluton. Assumpton 2. Tme-homogenety assumes tme homogeneous transton probabltes,.e. Q(=Q, W(=W and S(=S and a fxed recrutment polcy R(=R and r(=r. Furthermore, we assume that the desred personnel dstrbuton n ( t s fxed over tme. Defnton. The evoluton of the stoc vector s called stablzed n case there exsts a value t such that t : υ ( υ ( t. t = Remar that n case t t holds that ( t = υ ( t t to the matrx = M ( υ ( t. M υ, the sequence ( ( ( M υ converges for Assumpton 3. The evoluton of the stoc vector n ( s stablzed wth lm M ( υ ( = M. t Theorem 2. If Assumptons -3 are satsfed and lm n ( = n then t e n e = n I A R r.( I M (0 υ Proof. Snce the evoluton of the stoc vector n ( s stablzed wth lm M ( υ ( = M there exsts a υ ( t value t such that t t : n( = n( t. M n. I. A Rr (Theorem. In case the stoc vector n ( converges towards the lmtng dstrbuton n e, ths lmtng dstrbuton equals n e = n em n I A R r ne = n I υ υ A R r.( I M Accordng to Lemma the nverse matrx ( I M exsts.

12 Whch proves Theorem 2. Theorem 3. If Assumptons -3 are satsfed and all the egenvalues λ of M satsfy λ <, then Proof. lm n( t = n I υ A R r.( I M Accordng to Theorem there exsts a value t such that t t : n( = n( t. M n. I. A Rr υ (, snce the evoluton of the stoc vector n ( s stablzed wth lm M ( υ ( = M. For M, the nverse matrx t ( I M exsts accordng to Lemma. For n e = n I υ d ( = n( ne A Rr.( I M = n( t. M n. I. A Rr ne υ ( = [ d( t ne ]. M n. I. A Rr ne υ ( = d( t M ne M n I A R r ne υ = d ( t M and t > t, the dfference d( = n( ne satsfes: From the tme t the evoluton of the system s determned by the matrx M and therefore t > t : t t d( = d( t. M. Under the assumpton that all egenvalues λ of M satsfy λ < t holds that lm. M t t t = 0 and therefore: lm d( = 0. Consequently, lm n( t t = n e = n I υ Whch proves Theorem 3. A R r ( I M. 2

13 Theorem 4. If Assumptons -3 are satsfed, then all the egenvalues λ of M satsfy λ. Proof.. In case σ = { } no pull flows are nvolved and therefore the evoluton s determned by the non-negatve matrx M ( I W Q = P egenvalues λ of P satsfy λ [32]. = wth row sums less than. Consequently, the 2. Pull flows are comng n the model n case for some : ( > 0. Furthermore (accordng to (4: V ( > 0 n n ( t ( w > 0 V n n ( t < w As proved n Theorem 3, the stablzed evoluton of a stoc vector s governed by the powers t M of an MPP-matrx M. The propertes of the powers t M are determned by t λ, for λ egenvalues of M. In case the nvolved MPP-matrx M would have an egenvalue wth modulus greater than, the restrcton n n ( t < w can not be fulflled for all values of t and. Consequently, under the condton that the evoluton of the stoc vector s stablzed, for all the egenvalues λ of M holds that λ. Whch proves Theorem 4. Theorem 4 results n a necessarly condton to have an evoluton of the stoc vector n ( that s stablzed. Namely for an MMP-matrx ( υ ( M wth at least one egenvalue λ satsfyng λ > t s not possble to have value t such that t : υ ( υ ( t. t = 3

14 6. Numercal llustraton In ths secton, we provde a numercal llustraton of the results n ths paper. It concerns an organzaton n whch the manpower system s dvded n three homogeneous groups. The dentfcaton of the homogeneous groups s done based on the methodology n prevous wor [3-5]. Because of uncertan sales n the unpredctable organzatonal envronment, the company decdes to fx the worforce demand over tme as = ( n. The ntal stoc at t = 0 s gven by: n ( 0 = ( The fxed recrutment polcy s gven by. r = ( R. As shown n Bartholomew et al. [2], based on the estmator for transton probabltes gven by Anderson and Goodman [33], usng an hstorcal dataset, the tme-homogeneous wastage and push transton probabltes can be estmated. For the organzaton under study, the transton probabltes are gven by: P = We use equaton (2 to reformulate ths transton matrx P n terms of the parameters of the mxed push-pull model: Q = and I W = Notce that wastage s expected n every group of the system, such that Assumpton s satsfed. Further, the future personnel dynamcs depends on frms promoton polcy that regulates the pull transtons n the manpower system. To llustrate ths, we study two dfferent scenaros. 4

15 SCENARIO. Assume that, besdes the push transtons, the organzaton sets ts promoton polcy by a pull approach. In case the stoc n a personnel group s less than the desred number of employees, vacances arse whch are flled accordng to the promoton and recrutment polcy characterzed by: S = Whle vacances n group are entrely flled by external recrutments, vacances n group 2 and 3 are partly flled by promotons from respectvely group and 2. Furthermore, notce that the mxed push-pull model s fully determned by tme homogeneous transton probabltes. As a result, besdes Assumpton, also Assumpton 2 holds. Ths allows us to llustrate the theoretcal results of ths paper concernng the asymptotc behavor of the mxed push-pull model. For = 3, the personnel dynamcs n the mxed push-pull model can be regulated by eght possble transton matrces ( υ (. Gven Q, S and I W, by applyng (8, an overvew of M the matrces ( υ ( s gven n Table. It shows that all egenvalues of the matrces ( υ ( M M are real and between - and. Consequently, accordng to Theorem 2, n case the evoluton of the stoc vector s stablzed, the lmtng structures can be computed by (0. The resultng lmtng vectors ( υ ( are gathered n Table. n e By applyng (7, we compute the evoluton of the stocs. Table 2 shows the extrapolaton results, gven the ntal stoc vector ( 0 = ( n. In the frst and second tme nterval, by (6, the vacances n each group are computed. Snce V ( > 0 and V ( 2 > 0, we now that υ ( (2 = {,2,3 }. In tme perod [,3 = υ 2, personnel moblty s no longer regulated by pull transtons to group 2, snce 2 (3 = 0 υ ( =. Subsequently, as can be seen n Table 2, the evoluton of the stoc vector s stablzed, snce t 4 : υ ( = (4 = 3. Hence, snce V and 3 {,3 } υ Assumptons -3 hold, from Theorem 3, we now that the stoc converges to e ( 3 = ( n. As the results n Table 2 show, ndeed, for t 23 : n( = (

16 Table. Stablzed stoc evoluton for every possble transton matrx ( υ ( M υ ( M ( υ ( Egenvalues n e ( υ ( {,2,3 } λ = λ2 = λ 3 = 0.5 ( λ = λ2 = λ 3 = 0.33 ( λ = λ2 = λ 3 = λ = λ2 = λ 3 = 0.7 ( 243 ( {,2} λ = λ2 = λ 3 = 0.08 ( {,3 } λ = λ2 = λ 3 = 0. ( { 2,3} λ = λ2 = λ 3 = 0. ( λ = λ 2 = λ 3 = 0.7 (

17 Table 2. Extrapolaton Scenaro t n( V( υ ( {,2,3} {,2,3} {,3}

18 SCENARIO 2. Assume that the organzaton decdes not to use pull flows n ts personnel strategy. Analogue to scenaro, snce for the egenvalues λ holds that λ <, we now that f the evoluton of the stoc vector s stablzed, the lmtng stoc vector exsts (Theorem 3. Snce pull transtons are not possble n ths example, by defnton, the evoluton of the stoc vector s stablzed and υ (t =. By applyng equaton (0, the lmtng stoc s computed n Table and s gven by ( The extrapolaton results, computed by (3, are gven n Table 3. It shows ndeed that for t 82 : n( = ( Snce no pull transtons ntervene, scenaro 2 s n fact an llustraton of the tradtonal tmehomogeneous push model wth nown recrutment, for whch t s very well nown that.( I lm n( = R. r P t. Indeed, t s easly seen by (8 that M ( υ ( = [ I W ]Q.. Remar that accordng to (, n both scenaros and n every tme nterval, enough employees are avalable n the states for fllng the vacances by the proposed pull transtons, n order to meet the restrcton of the mxed push-pull model. 8

19 Table 3. Extrapolaton Scenaro 2 t n( V( υ (

20 7. Dscusson and conclusons In prevous wor on the tme-homogeneous mxed push-pull manpower model, the lmtng behavor was studed under the assumptons that push as well as pull flows are nvolved for ( each state and (2 each tme nterval. The man contrbuton of ths paper s that the asymptotc behavor of ths model s studed by relaxng both assumptons from prevous wor. In present wor, we studed manpower systems for whch the ntervenng transton mechansm can be dfferent accordng to the state. We examned the lmtng behavor for manpower systems havng a stablzed evoluton of the stoc vector. We showed that the evoluton of the stoc M vector can only be stablzed wth an MMP-matrx ( υ ( for whch each egenvalue λ satsfes λ. Moreover t s proved that n case λ < the stoc vector converges. An analytcal form for the lmtng stoc vector s found. Although ths paper maes a valuable contrbuton to the understandng of the asymptotc behavor of the mxed push-pull model, we need to consder some lmtatons that could be addressed n further research. Frst, we have shown that the manpower system converges f the evoluton of the stoc vector s stablzed and all egenvalues λ of the nvolved MPP-matrx have modulus λ <. We proved n ths paper that the evoluton of a stoc vector cannot be M stablzed wth a specfc matrx ( υ ( n case for at least one egenvalue λ holds that λ >. However, t s not yet nown what suffcent condtons the manpower system should satsfy n order to have a stablzed evoluton. The convergence has to be studed n the stuaton a stoc vector s stablzed and the nvolved MMP-matrx has some egenvalues λ wth λ =. Second, as seen n the second example, t s very well nown that for the tradtonal tmehomogeneous push model wth nown recrutment, the lmtng behavor s ndependent from the ntal stoc. It s not yet clear that ths s also the case n the mxed push-pull model. As the ntal stoc determnes whether or not n the early tme ntervals the pull mechansm ntervenes n one or more states, t has an mpact on the future personnel moblty. Hence, the ntal state strongly nfluences the early evoluton of the stocs. However, ths nfluence mght reduce and fnally dsappear as tme goes by. 20

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