Stochastic three-stage hiring model as a tandem queueing process with bulk arrivals and Erlang phase-type selection, M X /M (k,k) /1 M Y /E r /1

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1 Int. J. Mathematics in Operational Research, Vol. 5, No. 5, Stochastic three-stage hiring model as a tandem queueing process with bulk arrivals and Erlang phase-type selection, M X /M (k,k) / M Y /E r / Aliakbar Montazer Haghighi* and Dimitar P. Mishev Department of Mathematics, Prairie View A&M University, P.O. Box 59 MS 2225, Prairie View, Texas, USA amhaghighi@pvamu.edu dimichev@pvamu.edu *Corresponding author Abstract: We consider three possible stages for the handling of job applications in a hiring process as a network queuing model. Applications arrive at the first stage in batches of variable sizes according to Poisson process and are compiled in an A-Box. The batches of variable sizes (within a given minimum and maximum) are processed by a single-processor according to exponential distribution. The accepted portion of each processed batch moves to the second stage, the interview phase, and are piled in an I-Box. The interviews are conducted according to Erlang phase type; each phase according to exponential distribution. The successful applications are, then, directed to the third stage, final hiring phase, and are piled in an H-Box. Using decomposition of the system, we find generation functions and the mean of the number of applications in each of the first two stages. Explicit distributions of the number of applications are found for special cases and numerical examples are also provided. Keywords: hiring process; queueing model; network; Erlang phase type; Poisson; exponential; bulk; batch; single-processor; decomposition. Reference to this paper should be made as follows: Haghighi, A.M. and Mishev, D.P. (23) Stochastic three-stage hiring model as a tandem queueing process with bulk arrivals and Erlang phase-type selection, M X /M (k,k) / M Y /E r /, Int. J. Mathematics in Operational Research, Vol. 5, No. 5, pp Biographical notes: Aliakbar Montazer Haghighi is a Professor and Head of the Mathematics Department at Prairie View A&M University, Texas. He received his PhD in Probability and Statistics from Case Western Reserve University, Cleveland, Ohio, under supervision of Lajos Takács; and his BA and MA in Mathematics from San Francisco State University, California. His research publications are extensive; they include mathematics books written in Farsi and English. His latest books Queueing Models in Industry and Business (2nd edition, 22) with Mishev, as co-author, and Advanced Mathematics for Engineers with Applications in Stochastic Processes (revised edition, 2) with Lian and Mishev as co-authors, were published by Nova Science Publishers, New York. He is a life-time member of American Mathematical Society and Society for Industrial and Applied Mathematics; and the co-founder and Editor-in-Chief of Application and Applied Mathematics: An International Journal (AAM), Copyright 23 Inderscience Enterprises Ltd.

2 572 A.M. Haghighi and D.P. Mishev Dimitar P. Mishev is an Associate Professor in the Department of Mathematics at Prairie View A&M University, Prairie View, Texas. He received his MS and PhD in Mathematics from Sofia University, Sofia, Bulgaria, where he subsequently became Associate Professor and Head of the Department of Differential Equation. He has published many research papers in mathematics including one joint monograph with D.D. Bainov on differential equations, one with Dr. Haghighi on Queueing Models in Industry and Business in 28 and also Advanced Mathematics for Engineers with Applications in Stochastic Processes in 2, with Haghighi and Lian. Introduction In today s technologically oriented communication and manufacturing systems, the performance evaluation is possible by modelling them as queueing networks. Parametric decomposition is one of the most popular and effective methods in queueing network approximations. It is a parametric decomposition which decomposes a network into single nodes and the interactions between these nodes are captured by a few parameters. Due to the limitations of closed form solutions and exact algorithms to analyse general queueing networks, approximations are the tools researchers are focusing on. In most such queueing systems, the approximations are used, usually, on a few parameters such as the interarrival and service time distributions and the first two moments. Hence, large errors in many conditions are possible. Girish and Hu (996, 997a, 997b, 2) develop higher order approximations for the single server and tandem queues with infinite waiting buffer, general interarrival and service time distributions and focused on the phenomena of splitting, merging and feedback. Although, manufacturing and communication systems have only finite buffers, they expressed their hopes that their results lead to solutions for the finite buffer systems in the future. They illustrated the gains and the accuracy of these methods and offered the numerical examples. Using functional equation, Haghighi and Mishev (29) considered analysis of a two-node task-splitting feedback tandem queue with infinite buffers. Haghighi et al. (2) considered a single-server Poisson queueing system with splitting and delayed-batch-feedback, the case that the batch size is. They considered this system as a tandem queue. Algorithm to find solutions was offered. It is well known that Bailey (954) was the first to introduce bulk service. Bulk (or batch) service queues find applications in mass transportation. We, in this paper, take it from a managerial view point for a hiring process. Neuts (967) studied a general class of bulk queues with Poisson input. He defined different cased of bulk services and analysed the general case. Kleinrock (975) considered usual bulk service models that are equivalent to Erlang arrival systems. He studied variety of cases such as parallel and parallel-series services. Madan et al. (23) studied two M X /M a,b / models with random breakdowns. They also gave the generating function of the queue length for the case with no breakdowns. Jain et al. (27) considered the general bulk rule with general service distribution. Among other things, they gave the generating function of the number in the system at the departure of the n th customer. Arumuganathan et al. (28) discussed the steady-state analysis of a non-markovian bulk queueing system with N-policy and different types of vacation. Sikdar et al. (28) have discussed a batch arrival batch service queue with finite buffer under with vacations. Khalaf et al. (2) studied an

3 Stochastic three-stage hiring model as a tandem queueing process 573 M [X] /G/ queue with Bernoulli schedule, general vacation times, random breakdowns, general delay times and general repair times. They assumed that the repair process does not start immediately after a breakdown and there is a delay time waiting for repairs to start. They found the steady state solution of the system using supplementary variables technique. Haridass and Arumuganathan (2) considered the bulk service queueing system with variant threshold policy for secondary jobs. The derived the queue size distribution, steady-state condition and various performance measures. Additionally, they have developed a cost model for the system. Laxmi and Yesuf (2) considered an infinite buffer single server batch service queue with single exponential working vacation policy. They used supplementary variable technique and the recursive method to develop the steady-state queue length distributions at pre-arrival and arbitrary epochs. They obtained some performance measures and numerical results. Gupta and Banerjee (2) considered a Markovian queue with finite buffer and bulk-service with general distribution. Among other things, they found the steady-state joint distribution of the number of tasks in the system and number with the departing batch at the departure epoch. This study was a variation of an earlier study by Gold and Tran-Gia (993). Singh et al. (2) investigated a single server bulk queueing system with state-dependent rates and second optional service. By using supplementary variable technique, they found the probability-generating functions of the queue length distribution. Additionally, they have provided the numerical examples to verify the tractability of performance measures. Choudhury and Medhi (2) considered a multi server Markovian queuing system, where the customers may balk as well as renege. They provided explicit closed form expressions. They have provided a numerical problem with design aspects to demonstrate results derived. Singh et al. (22) studied single-server state-dependent queuing systems, wherein tasks arrive in batches and follow the Poisson process with state dependent arrival rates. By using supplementary variable techniques, they provide the probability generating function of the queue length distribution and consider various performance measures. They also provided numerical examples to verify the tractability of performance measures obtained analytically. The remaining parts of the paper are organised as follows: Section 2 illustrates the model; Section 3 is devoted to analysis of the first stage (referred to as the FS); Section 4 discusses the analysis of the second stage (referred to as the SS); and finally, Section 5 includes discussions regarding the third stage (referred to as the TS), concluding remarks and future works to do. 2 The model Consider personnel hiring policy at a large company such as IBM, AT&T, Dell or Exxon Mobil. Due to turnover and retirement of employees, there is a constant need for hiring new employees. Hence, it is assumed that hiring process is a continuous task, arrival of applications is non-stop, and applications are accepted at any time electronically only through the company s website. It is also assumed that the company s website is capable of receiving a group of applications at the same time. It should be noted that in this paper the words bulk, batch and group will be used interchangeably.

4 574 A.M. Haghighi and D.P. Mishev There are three stages for the hiring process. The FS is the initial processing of arrived applications from a box, called the A-Box. This is to make sure that an application meets the minimum qualifications. It is assumed that applications arrive in batches with variable sizes according to a time-homogeneous Poisson process and are stored as groups in the A-Box. Assuming the A-Box to be finite in capacity will be led a model of a tandem queue with blocking. This type model might be more realistic and that is why recently there has been an increase interest in such a model due to their wide applicability [see, Haghighi and Mishev (27), for example], in this paper, we design our model with infinite buffers to avoid too much complication involved in such a case. Thus, on the one hand, we are modelling a real-life problem using a queueing network, we are replacing a complex real-life model with the one we are to consider, which is very close to it, yet, structurally, much simpler. On the other hand, since, generally, we will not be able to find explicit formula for distributions we are looking for, to justify the choice of model, it is not sufficient to show that the traffic density is less than. Hence, we would choose a tolerable bound for the error of our approximation and specify conditions of stability of the network. There is an automatic single-processor (scanner) assigned to this stage that scans applications by batches of variable size from the A-Box. Processing of a batch in this stage follows an exponential distribution. However, although processing of batches is in the order of their arrivals, no ordering within a batch. The SS is a stage of series of interviews. Applications that have undergone the initial review and posses the minimum qualifications will move on to the SS for interview and the rest will be discarded. That is, only a portion of each batch will continue undergo the rest of the hiring process. Each portion by itself is a batch and of variable size. These batches when arrived to the SS will be held, in order they arrive, in a new infinite capacity folder called I-Box. Ordering within a batch is random. Thus, application batches are lined up only in order of their arrivals and will move from within a batch out to the I-Box one at a time. If a batch arrives to find no applicant is being interviewed, an applicant from the batch will be called immediately and go through the interview phases. Each applicant related to an application in the I-Box is assumed to go through a number of interviews in sequence before leaving the SS. And the TS is designated to complete the hiring process of the selected candidates, such as background checks and exchange of contracts for signatures. After completion of interviews at the SS, application of applicants who pass the series of interviews successfully will be accepted to be hired. Applications of hired applicants will be filed in a box called H-Box. In the next sections we analyse the first two stages. Since we are assuming continuous hiring, it would mean acceptance of applications as they arrive. We could have limited the number of available positions to a finite number, i.e., H-Box would be finite. In that case, as soon as the H-Box is full, the system will lose all its non-complete applications and the potential arrivals. The system, then, may be considered as a reneging (by design) queueing system. Due to the length of present paper, detail of this case will be included in an upcoming paper. Hence, there is no need for further discussion of the TS in this paper.

5 Stochastic three-stage hiring model as a tandem queueing process The analysis of the FS, initial processing As mentioned in the previous section, applications arrive in batches according to a time-homogeneous Poisson process with parameter λ and will be stored as groups in the A-Box. Let the batch size denoted by X, with probability distribution function PX { = x} = α, x=,2,. (3.) x That is, the probability that a batch of size x applications arrive in an infinitesimal interval of time (t, t + Δt) is λα x Δt + o(δt) and of no arrival is λδt + o(δt). We denote the mean random size of a batch by α. Then, the arrival process in this case will be a compound Poisson process with arrival rate λα x, α x, α, x= x = and mean arrival rate λα. We assume that both mean and variance of X are positive and finite. The applications from the A-Box are processed with a minimum of k applications and maximum of K (K k ). This is the general bulk processing (GBP) rule of Neuts (967). Processing of a batch follows an exponential distribution with parameter ν, i.e., with mean /ν. Hence, the FS becomes an M X /M (k,k) / queueing system with the utilisation factor λα ρ =. (3.2) Kν The way GBP works is as follows: suppose that after completion of processing a batch of size, say b, (k b K), q, q, applications will remain in the A-Box. Then, there will be three cases: Case If q < k, then the next processing will not start until the number of applications reaches the minimum number k. Case 2 If k q < K, then the batch processor will pick up q applications and starts processing. Case 3 If q K, then the processor will pick up K applications, in order of their arrivals, and starts processing. The rest of applications will remain in the A-Box for the next round of processing. Note that, although, in some cases, delaying the processing by imposing a minimum, k, may seem a waste for the processor s time to remain idle until the number of applications reaches k, and, thus, costly, with today s technology and electronic processing, the processor may be busy doing other things while k is to be achieved. For instance, consider the example in Neuts (967) for explanation. Note also that the FS is a non-birth-death process, since the transitions, in case of bulk arrival of size x(x ), will occur to a state different by x and will not necessarily be a neighbouring state. We define the state of the FS, j, as the number of applications in the A-Box and being processed at an epoch of time. We denote by the probability that at time P j (t) the FS is in state j. We also denote by P j, P j = lim t P j (t), the stationary probabilities of P j (t). The system of balance difference equations for the stationary distribution of the number of applications in the FS, in case of M X /M (k,k) /, is as follows:

6 576 A.M. Haghighi and D.P. Mishev K λp = ν P, i= k i n K λpn = λ α, if, i ipn i + ν P i k n i n k = + = n K ( λ+ ν) Pn = λ α, if. i ipn i + ν P i k n+ i n k = = (3.3) This is because, for balancing the flow of applications, if there are i applications in the FS and it is to be empty, the only possibility is that batches of sizes k up to K be processed with rate ν. On the other hand, to have n applications in the FS, either the FS is i applications short or i applications over. In the first case there should be flow in of applications with variable sizes or be processes, in the second case with rate ν. To solve system (3.3), as standard, we define the probability generating functions of α x and Pj, respectively, as follows: with x j α x j (3.4) x= j= Az ( ) = z and Gz ( ) = Pz, z <, A() =, A () = α and G() =. (3.5) Hence, multiplying both sides of the second and third equations of (3.3) by z n+k and sum over n, multiply the first equation of (3.3) by z K, add, noting that and n n αipn iz = A( z) G( z), (3.6) n= i= = ( ), K K K j n+ K K i K j i Pn+ iz G z z z Pz i n= i= k i= k j= k i= we will have (3.7) K k n K K j j i n ν n= j= k i= i K K K i [ λ λaz ( ) + ν] z ν z i= k νz P z z Pz Gz ( ) =, (3.8) which coincides with formula (23) of Madan (23), i.e., M X /M (k,k) / system without server breakdowns. Finite sums in the numerator of (3.8) may be found using the standard argument of applying Rouche s theorem. That is, knowing A(z), the number of zeros of denominator of (3.8) within and on the unit circle is the same as the ones of the numerator. Thus, the unknown probabilities P j, j =, 2,, K would be found, and G(z) will be completely determined.

7 Stochastic three-stage hiring model as a tandem queueing process Special case, k = K = If k = K =, then (3.8) reduces to the probability generating function for the M X /M/, which is Gz ν ( zp ) ( ) =, [ ] ν( z) λz A( z) (3.9) as in Medhi (23). From (3.5) and applying L Hôpital s rule on (3.9), we will have ν P lim Gz ( ) = z ν λ [ Az ( ) za ( z) ] νp νp ν λa () ν + λα = = = z=. (3.) Hence, from (3.) and (3.2) for K =, we will have P = ρ. (3.) Thus, substituting (3.) in (3.9) we obtain ρ. Gz ( ) =. λz [ A( z) ] ν ( z) i () i () (3.2) Letting E( X ) = α, i 2, α = α, and N the number of applications in the FS, then the mean and variance of N can be obtained using (3.2), respectively, as follows: and (2) ρ α + α =, ( ) E N Var N ρ 2α (3.3) ρ 2α (3) + 3α (2) + α 2 = +. ρ 6α (3.4) ( ) E ( N ) If, for instance, applications arrive in batches of size or 2 according to Bernoulli distribution such that α = p, α 2 = p, < p <, and α x =, for x 3, then ρ = (2 p)λ/ν, ρ < and A(z) = pz( z) + z 2. From (3.2), the generating function of the number of applications in the FS will be given by ρ Gz ( ) =. λ λp 2 z z ν ν (3.5) Expanding (3.5) yields distribution of the number of applications in the FS, as follows:

8 578 A.M. Haghighi and D.P. Mishev ρ ρ ρ Pn = 2 ρ + 4(2 ρ)( ρ) 4 2ρ ρ + 4(2 ρ)( ρ) + ρ n+, n+ n ρ + 4(2 ρ)( ρ) + ( ) ρ n =,,2,. n (3.6) From (3.3) and (3.4), the mean and variance of the number of applications in the FS, in this case, can easily be calculated, respectively, as ( ) E N (3 2 p) ρ = (2 p)( ρ) (3.7) and 2 ρ (3 2 p) ρ Var ( N ) = 5 4 p +. (3.8) (2 p)( ρ) (2 p)( ρ) If p = /2, then ρ = 3λ/2ν, and from (3.6) to (3.8), we will, respectively, have the distribution, mean and variance of the number of applications in the FS as ρ ρ Pn = ( ρ ) 2 3 ρ + 3 n n+ n+ ρ 3 ρ , n =,, 2,, ρ ρ ( ) E N 4ρ =, 3( ρ) (3.9) (3.2) and ( ) Var N ρ(2 + 9 ρ) =. 9( ρ) 2 (3.2) As another example for the case M X /M/, let X have geometric distribution, i.e., x α x = PX ( = x) = p( p), < p<, x=,2,. (3.22) Then, Medhi (23) shows that the generating function, expected value and the distribution are, respectively, as pz Az ( ) =, (3.23) ( pz )

9 Stochastic three-stage hiring model as a tandem queueing process 579 EX ( ) = α = A () =. (3.24) p and ρp( ρ) P =, n, n [ p( ρ) ] n (3.25) with and P = ρ and ρ = λ/pν. Jensen et al. (977) have studied M X /M/. They assumed arrivals by bulk of size j, j =, 2,, m, according to a Poisson process {N j (t) : t } with mean λ j, j =, 2,, m, and that these Poisson processes are mutually independent. Under these assumptions, letting M(l) = N (l) + N 2 (l) + + N m (t), the superposed process {M(l) : t } is a m Poisson process with mean λ. j j = λ M(t) represents the number of batch arrivals in = (, t] and mean rate of batches as λ. Denoting P(X = j) = λ j /λ, j =, 2,, m, they took X to have a multinomial distribution. Assuming exponential single-server service with mean service time /μ, for m = 2, they conjectured that [ n/2] i n 2i ρρ 2 i i= Pn = P n, in, 2i n, (3.26) where [.] is the greatest integer symbol, ρ j = jλ j / μ, j =,2, Po = ρ, ρ <, 2 ρ ρj. j= 3.2 Number of batches in the FS at a departure epoch Back to the general case, M X /M (k,k) /, let us denote by D n the number of batches of application in the FS at the departure of the n th batch from the FS and D the initial number of application batches waiting to be processed, i.e., there are D applications in A-Box when the screening of applications begins. It is easy to see that the departure point process {D n : n } forms a Markov chain which is embedded over the non-markovian process at regeneration points such as process completion epochs of batches. Hence, we will consider the process at such points. Let ξ n denote the number of arrival batches during processing time of the n th batch, that are iid random variables, and d P( ξ = j), j =,,. (3.27) j n At a departure point, the elapsed processing-time for the entering batch of applications is zero. Hence, the number of applications at a departure epoch is dependent upon the number at the previous departure epoch and the number of arrivals during the processing time. Now, if the n th departure batch does not leave any application behind at the time of its departure, then, the next batch will leave behind only those arrived during its processing time. On the other hand, if the n th batch leaves some behind at time of its departure, then the next departing batch will leave those minus the number processed plus the number that arrived during its processing time. Hence, based on the three cases

10 58 A.M. Haghighi and D.P. Mishev mentioned above, let q be the number of applications left behind at the departure of the nth batch. Then, we have ξ n+, if Dn =, Dn+ = Dn q+ ξn+, if Dn >, k q < K, Dn K + ξn+, if Dn >, q K. (3.28) Now since arrival of applications is a Poisson process with parameter λα x, α x, the conditional distribution of ξ n given the processing time Θ = θ will be ( λα θ ) j λα θ x x e P( ξn = j Θ= θ) =, αx, j =,, 2,. (3.29) j! Thus, ( ) ( ) ( λα θ ) x dj = P ξn = j = e νe dθ j! j x j λα θ νθ ν λα x j ( λα x + ν ) θ = θ e dθ, αx, j,, 2,. j! = (3.3) Denoting the transition probabilities of the Markov chain {D n : n } by p ij = P(D n+ = j D n = i), from (3.28) and (3.3) we will have d j + i K, if i > K, j i K, pij = d j, if i K, j,, if i K, j < i K. The transition matrix of (3.3), denoted by P, is as follows: (3.3) P d d d2 d3 d d d2 d 3. = d d d2 d3 d d d2 d d (3.32) The matrix in (3.32) is known as of the upper Hessenberg form. Since the first K + rows of P are identical, it indicates that the Markov chain is irreducible and aperiodic. If we denote by n the steady-state probability of n batches of applications in the FS at a departure epoch, then from (3.2), if the traffic intensity ρ = λα / Kν <, it can be shown that P is ergotic and { n : n } exist, see Jain et at. (27). The system of balance equations, in this case, is K K+ j (3.33) Π d Π + d Π, j =,,. j j i j + i K i i= i= K

11 Stochastic three-stage hiring model as a tandem queueing process 58 with Π. j= j = To solve (3.33), we may go through the standard method of generating function or use of the Neuts geometric matrix method. For the generating function method, we define the generating function of (3.33), denoted by (z), as K K+ j j j i j j + i K i i= j= j= i= K (3.34) Π ( z) = Π d z + z d Π, With () =. Note that since (z) is a probability generating function, it is absolutely convergent for z. From (3.4), (3.33) and (3.34), we will have K i ( z z ) K i= i Π ( z) = Π K z Δ( z), if z, (3.35) j where Δ ( z) = d. j j z = λα Note that Δ() = and Δ () = ρ =. Note also that { n : n } exist for ρ < K. ν Using standard argument of Rouché s theorem and L Hôpital s rule, if, r, r 2,, r K are roots of K z Δ ( z) =, (3.36) inside and on the unit circle, then (3.35) can be rewritten as K z ri ( K ρ)( z ) i= ri λα Π ( z) =, if z, ρ = <. K z Δ( z) Kν (3.37) Note that if K =, then (3.37) becomes ( z ) Δ( z) Π ( z) = ( ρ), z Δ( z) (3.38) which is the Polaczek-Khinchin formula. To utilise (3.37), we need to find roots, r i s, of (3.36). To find r i s in (3.37), we have to find roots of the equation (3.36) or equivalently of the function where K f ( z) = z Δ ( z), (3.39) j= j Δ ( z) = d z, inside the unit circle, in addition to, which is on the unit j circle. Note that Δ(z) is a function of z and x since although it is the generating function of d j, d j is a function of x.

12 582 A.M. Haghighi and D.P. Mishev Müller method is the one usually used to approximate the roots of (3.39), particularly, in queueing models. Steps to apply this method may be found in Jain et al. (27). However, QROOT-Software of Chaudhry (993) is also available. We in this paper use geometric matrix method of Neuts that follows. We denote the coefficient matrix of (3.33) by Q, which is d d d d d Q =. d2 d2 d d Note that that Q is column stochastic, i.e., for each column that the first K elements of each row are the same. Let be a column vector of unknowns j, j =, 2,,, i.e., 2 T j= (3.4) j, d =., Also, note Π= Π, Π, Π,. (3.4) Then, (3.33) can be rewritten as QΠ=Π, and Π j =. (3.42) j= The matrix Q, except for its first K columns, is a Toeplitz matrix. Toeplitz matrices model a wide variety of problems in queueing theory. However, the advanced algorithmic tools developed in the field of Toeplitz computations are not generally used in the solution of queueing problems. Bini and Meini (996) make a survey of some recent results encountered in Toeplitz matrix analysis that devise very efficient algorithms for solving a wide class of queueing problems. We choose some of these results and one from Ramaswami (988a) formula to offer an algorithmic solution for the problem under consideration. j 3.3 Algorithmic solution for (3.33) We partition Q into (K ) (K ) blocks and obtain a block matrix, denoted by Q B, as follows: Q B B A B2 A A =, B3 A2 A A (3.43) where

13 Stochastic three-stage hiring model as a tandem queueing process 583 and d d d A = d2 d d, dk 2 dk 3 d d[ i( K )] d[ i( K ) ] d[ i( K ) ( K 2)] d[ i( K ) ] d[ i( K )] d [ i( K ) ( K 3)] A + i =, i =,2, d[( i K ) + K 2] d[( i K ) + K 3] d[( i K )] d[( i )( K )] d[( i )( K )] d[( i )( K )] d[( i )( K ) ] d[( i )( K ) ] d [( i )( K ) ] B i =, i =,2,. d[( i )( K ) + K 2] d[( i )( K ) + K 2] d[( i )( K ) + K 2] (3.44) (3.45) (3.46) T T T Let the matrix, given by (3.4), Π = Π, Π, Π2,, be partitioned as follows: Π = Π, Π, Π2,, ΠK 2 Π = ΠK, ΠK, ΠK+,, Π2K 3 Π 2 = Π2( K ), Π2K, Π2K,, Π3K 4 Π = Π, Π, Π,, Π. n n( K ) n( K ) + n( K+ ) + 2 ( n+ )( K ) We now continue solution of (3.33) via the following steps. The idea is to find a (K ) (K ) matrix G given by the minimal nonnegative solution of the non-linear matrix equation G = G i A i= i (see Neuts, 989; Ramaswami, 988b). Then, we use matrix G and the Ramaswami formula (see Ramaswami, 988a) to find, by recursive T computations, components of the vector Π= Π, Π, Π2,. To do that, we let R j be an approximation for G and take the following steps: Step 3. Choose data. Test for ρ <. Step 3.2 Choose a distribution for X, i.e., define α x. Step 3.3 Calculate the matrix d j given in (3.3). Step 3.4 Write matrices A, A i and B i. Note that in programming for infinite matrices A i and B i, they are approximated by finite matrices. Hence, the maximum value, replacing

14 584 A.M. Haghighi and D.P. Mishev infinity, of j in d j in Step 3.3, should be at least (K )( + Y), where Y is the valued replacing infinity for values of i in defining Ai and Bi. Step 3.5 Let C = (I A ) and D = A C. Choose R =, where is a (K ) (K ) zero matrix. Assume that R =. Let i ( i= 2 ) R = D + R A C, m =,, 2,. m+ m i Step 3.6 Check if for a desired ε, max i, j (R m+ R m) ij < ε, m =,, 2,. If Step 3.6 holds, then write G R m+. Otherwise, repeat Step 3.5. Test to make sure all eigenvalues of G are within the unit circle. Step 3.7 Compute j i 2 i j i i+ i+ 2 j= i A = G A = A + GA + G A +, i =,2,. and j i 2 i j i i+ i+ 2 j= i B = G B = B + GB + G B +, i =,2,. Step 3.8 Solve equation MX =, where M = B I,X =, X,X = X, X,, X. 2 K 2 as follows: Step 3.8a Delete the first row of M and rename the remaining M. Step 3.8b Delete the first column of M and call the remaining as M 2. Step 3.8c Choose the first column of M, multiply it by ( ) and call it M 3. Step 3.8d Write X = M M. 2 3 Step 3.9 Find L = (I A ). Step 3. From Step 3.9, find X i as a matrix product i T T Xi = LBiX + Ai+ jx j, i, j= or T T T T Xi = L B X A X A X2 A2X i+ + i + i + + i, i.

15 Stochastic three-stage hiring model as a tandem queueing process 585 Step 3. From Step 3., find S = i= X T. i Step 3.2 From Step 3. and 3., write Π i = X. i S Note that Π i is the vector of unknown probabilities i, each vector has K component. Step 3.3 From Step 3.2, write Π= Π, Π, Π2,. Note that the size of this vector in numerical case would be (K ) (finite ) Example 3.3.: We take the following data: λ = 4, ν = 7, K = 4. We assume that X has a geometric distribution as in (3.22) with parameter p. Let x λ p =.2. Hence, α x =.2(.2), x =,2,, α = / p = 5, ρ = =.743 < Kpν and A(z) =.2z/(.8z) In the computation, we denote the maximum value of x by X, and let it equal to. Utilising data given above, the exact value of d j is: j ( ) 7 x 4.2 (.2) j ( x ) d = θ j θ e 4.2 (.2) 7 dθ, j! + x =, 2,, j =,2,. To approximating d j, we let the maximum value for j, denoted by J, for infinity, be [(K ) ( + max of i)]. Hence, utilising the data given above, J = 2. d j, in this case is a by 48 matrix, whose elements are zero starting from column 8. Thus, with values of x, as rows, and 8 values of j, as columns, an approximation of d j is listed in Table. In computing matrices A i and B i, we let the infinity (maximum value for i) equal to 3, since values for greater than 3 are about zero. A, A i, B i and G are each a matrix. G in this case is also a matrix as: G = Values of X, X, in X 2 this case are listed in Table 2. T Finally, in this case we have as Π = Π, Π, Π 2 : T T Π = ,

16 586 A.M. Haghighi and D.P. Mishev Table d j for the Example

17 Stochastic three-stage hiring model as a tandem queueing process 587 Table 2 Values of X, X, and X 2 for the numerical example X X X (.e-3) (.e-3) (.e-3) (.e-4) (.e-4) (.e-4) (.e-4) (.e-5) (.e-5)

18 588 A.M. Haghighi and D.P. Mishev 4 Analysis of the SS, interview processing Applications that undergone the initial review and posses the minimum qualifications will move on to the SS for interview and the rest will be discarded. That is, only a portion of each batch will continue undergo the rest of the hiring process. Each portion by itself is a batch and of variable size. These batches when arrived to the SS will be held, in order they arrive, in a new infinite capacity folder called I-Box. Ordering within a batch is random. Thus, application batches are lined up only in order of their arrivals and will move from within a batch out to the I-Box one at a time. If a batch arrives to find no applicant is being interviewed, an applicant from the batch will be called immediately and go through the interview phases. If all information we have regarding arrival of applications from the FS to the second are that a portion of each batch of applications passing through the FS will move on to the SS and that the times between arrivals of these portions to the SS are independent and identically distributed random variables, then arrivals to the SS may be described, in Kendal notation, as GI Y. Each applicant related to an application in the I-Box is assumed to go through a number of interviews in sequence before leaving the SS. We consider an interview as a phase and all phases as part of the interview process for an applicant in the SS. These phases are real interviews versus possible imaginary ones as originally studied by Erlang. Originally, Erlang introduced the phase type technique. However, long later, Gaver (954) extended Erlang s method by introducing number of phases to be potentially infinite and that a customer may demand j phases of service at its arrival with probability c j. Later, Luchak (956) and Wishart (959) discussed Gaver s idea. In this connection, see Chaudhry and Templeton (983). Jaiswal (96) modified the method used by both Gaver (954) and Luchak (956) introduced the Erlang s phase process with a vector process {N(t), R(t)} as the process for length of the system, where, in our case, N(t) denotes the number of applications and R(t) denotes the number of phases of interview, in the SS, that remain to be completed by the application currently being processed at time epoch t. We note that N(t) could be considered as the system-length that includes an application currently in process or the queue-length that contains only applications in the I-Box. As in the Erlang s method, the number of phases may be counted backward, i.e., an application that has entered phase r( r < ) with probability c r will be processed an exponential time and then will move to phase (r ) and so on through phase before moves on for the next stage or exits the system. Now, depending upon the nature of a particular position an applicant is applying for, the number of interviews may vary and, thus, may be a random variable. We suppose that each applicant will go through the same r phases of interview. Duration of each interview is assumed to be exponential. Assuming each phase being conducted by a single interviewer makes the SS as an GI Y /E r / queueing system. See Figure.

19 Stochastic three-stage hiring model as a tandem queueing process 589 Figure SS, an Erlang r-phased processing system, GI Y /E r / While the system GI r /M/ and M r /E r / have been extensively studied, the system GI Y /E r / has not. In fact, Chaudhry and Templeton (983) have a section of a chapter on this latter topic. Wishart (959) analysed the system GI/E r /with the aid of an embedded Markov chain. He derived the stationary distribution for the number of customers in the system at epoch of arrival. He also found the stationary distribution of the waiting time for an arbitrary customer. Let Y be a random variable representing the size of batches arriving to the SS with probability distribution function with PY { = y} = σ, y=,2,, (4.) y= y σ y =, and the mean random size of a batch σ. Now, since initial reviewing of applications in the FS follows an exponential distribution with parameter ν, for the stationary process, as long as the FS is busy (which was assumed for our model), interdeparture distribution will be exponential with parameter ν. Hence, arrival to the I-Box follows a Poisson distribution with parameter νσ. denoting it by I( y, νσ ), we have y ( νσ ) νσ y I( y, νσ ) = e, y =,,. (4.2) ( νσ )! Hence, what (4.) means is that the probability that a batch of size y applications arrive in an infinitesimal interval of time (t, t + Δt) is νσ y Δt + o(δt) and of no arrival is νδt + o(δt). Then, the arrival process to the I-Box will be a compound Poisson process with arrival rate νσ y, σ y, and mean arrival rate νσ. We assume that both mean and variance of Y are positive and finite. For the average time of an application from the I-Box that has to go through all phases, Chaudhry and Templeton (983) have the following: Let the number of phases be a random variable, say, R, with a maximum of M possible phases. Let (4.3) M M r r= r r= r PR ( = r ) = c, r M, with c = and c = rc. Then, processing time of applications in the SS will be iid random variables having modified Erlang distribution function with parameter μ, say, H(t) with density function h(t) as

20 59 A.M. Haghighi and D.P. Mishev M r= r μμt μt r dh () t ( ) ht () = = c e, t. dt > (4.4) ( r )! From (4.3) and (4.4), we will have the average processing time of applications in the SS as c h = = < μ tdh() t. (4.5) It is clear that the time-dependent process {N(t)} of the number of applications in the SS is non-markovian. However, it can be made Markovian by considering the vector (N(t), R), where R is the random variable representing the number of phases left to be completed by the application being processes. Hence, we can consider the state of the SS as (), (, r), (2, r),, r M. Let us denote and Pnr, () t = P{ N() t = n, R = r}, n >, r M (4.6) P () t = P { N () t = }. (4.7) Of course, both probabilities in (4.6) and (4.7) depend upon the initial value N(). Thus, transient system of differential-difference equations (or the forward Kolmogorov equations) for the number of applications jointly with the number of phases left to be completed may be written. * Let Pnr, () s denote the Laplace transform of P n,r (t). We also define the following generating functions M Cu ( ) = cu, r= r= n= * n, n= r r M * n, r G(,;) zus = P () szu, r G (;) z s = P () s z z <. n n r (4.8) Then, using (4.8) and standard argument, we obtain Cu ( ) * μ G ( z; s) C( y) Gr (,;) z u s =, s + ν + μ νaz ( ) n where { ν } * P () s [ A() z ] s + z Gr (;) z s =, μ C( y) z (4.9) (4.)

21 Stochastic three-stage hiring model as a tandem queueing process 59 l z () =, s+ ν A( z ) P s (4.) μ y, (4.2) s + ν + μ νaz ( ) and z is the only root, inside the unit circle, of denominator of the right hand side of (4.9). Hence, distribution of the number of applications in the SS can be obtained from (4.9) to (4.2). 4. Distribution of number of phases and applications in SS Conolly (96) has considered the limiting behaviour of system GI r /M/ in continuous time; Takács (962) discussed the transient behaviour of the number of customers in the system and gave an implicit form of the distribution. Takács (962), Conolly (96) and Prabhu (965) addressed the equivalence of the busy period for the two systems GI r /E/ and GI/E r /. Chaudhry and Templeton (983) showed that the number of customers, N n, in system GI r /E/ has the same distribution of the number of the phases to be completed by the applications in the SS just before arrival instant of the n th application, Φ n, in the system GI/E r /. Hence, the number of applications in GI r /E/ system may be identified with number of the phases in system such that distribution of service-time of a batch in GI r /E/ is the same as the service time distribution of an application in the system. We now want to find distribution of the number of phases in the SS. Total time spent for interview (phases) at this stage is the sum of r iid exponential interview periods each having parameter rμ. Each one of the exponential periods is a phase and a processing in this stage is completed when an applicant goes through all interviews. Hence, the time, τ r, spent in the r th phase is independent and identically distributed according to negative-exponential distribution with mean /rμ so that P{τ r > t} = e rμt Thus, the total interview time at this stage τ =τ + τ τ r, has a scaled-modified χ 2 (chi squared) distribution with mean /μ and 2r degrees of freedom, i.e., Erlangian Er or gamma distribution with integer-valued first parameter (r, rμ). Hence, the processing probability density function at this stage, denoted by h(t; r, rμ), is r dh () t rμ( rμt) rμt htrr (;, μ) = = e, t. dt ( r )! (4.3) As we mentioned before, since the Erlangian phases are in reverse order, the system state at an instant is considered in terms of phases remaining for the process to be completed. That is, if an application is in the r th phase, it has yet to go though M (r ) phases of process. Hence, if at any instant there are j, j >, applications in the SS and the application being processed is in the r th phase, r M of the process, then the total number of phases to be completed by the applications in the SS is jm r +, j >, < r < M and the system is in state jm r +. This is because each application must go through r phases. Since there is one application being processed, there remains j applications, i.e., M(j ) phases. The one in process is in phase r, i.e., M (r ) phases left. The total number of phases in the system, therefore, is the sum of these two quantities.

22 592 A.M. Haghighi and D.P. Mishev Now, let us denote by Φ(t) the number of phases to be completed in the SS at the time t. Let us also define q () t P{ Φ () t = n}, (4.4) n with the stationary probability q n, i.e., q n = lim t P(Φ(t) = n), i.e., q n is the probability that the system (SS) is in state r. Then, the system of balance equations for the number of phases of the n th applicant in the SS to be completed is as follows: νq = rμq, ( ν + rμ) q = rμq + νσ q, if n =, 2,, n n n+ i= n i i (4.5) with q. n= n = We will give an algorithm to solve system (4.5) recursively. However, to solve it using probability generating function (pgf) of q n, similar to (3.4) and (3.5), we define probability generating functions (pgf) as follows n y y n y= n= (4.6) Sz ( ) = σ z and qz ( ) = qz, z <, with q() =, S() =, and S () = σ. (4.7) Going through the standard pgf method, interchanging the order of summations for the double sum n = n= i= i= n= i+ and regrouping terms, we have rμ ( ν + rμ) q( z) q = q( z) q q z + νq( z) S( z). z (4.8) Now, using the first equation of (4.6), (4.8) and some algebraic manipulations, we will have qz rμ( z) q ru( z) ν z( S( z)) ( ) =. Dividing both sides of (4.9) by rμ( z) and using L Hôpital s rule, we obtain q and, hence (4.9) = ρσ (4.2) rμ( z)( ρσ) qz ( ) =. ru( z) ν z[ S( z)] (4.2) From (4.2) we have E[Φ(t)] = q () and to find it, we can rewrite (4.2) as

23 Stochastic three-stage hiring model as a tandem queueing process 593 μσ z[ S( z)] qz ( ) =, where gz ( ) =. ρ gz ( ) z (4.22) Note that lim z gz ( ) = S () = σ and lim z g ( z) = ( σ + S () / 2). Thus, denote the mean number of phases in SS by L SS, we will have S () ρσ + 2 LSS = E[ Φ ( t)] = lim g ( z) =. (4.23) z ρσ Denote by Q n the distribution of number of applications in SS. To have n applications in SS, there are n waiting and one is going through phases of an interview. Each of the n waiting application has to go through r phases and the one in the interview process may have left through r phases yet, to make the total possible of phases equal rn. Hence, the relation between the number of phases and number applications in SS is with nr Qn = qi, n (4.24) i= ( n ) r+ Q = q = ρσ. (4.25) We now offer three examples of distribution for Y and find their mean of Φ(t) for each distribution to be used in our numerical example later. Example 4..: Let us assume that after a batch is processed in the FS, only j, j x, applications are selected and sent to the SS. Then, S(z) = z j. In this case, (4.9) becomes rμ( z)( ρ) ν qz ( ) =, ρ =. j+ rμ + νz ( ν + rμ) z rμ From (4.26), we have (4.26) L SS ρ j( + j) ν =, ρ =. 2( ρ j) rμ (4.27) Example 4..2: If we assume That size of portions from each batch in the FS that is accepted to move on to the SS is either with probability p or 2 with probability p, i.e., σ y = P(Y = y) such that σ = p σ2 = p, σn =, for all other values of n. (4.28) Then, S(z) is 2 2 S( z) = σ z+ σ z, (4.29) and, then (4.9) becomes rμ( z)( ρ) qz ( ) =. 2 rμ( z) νz pz ( p) z (4.3)

24 594 A.M. Haghighi and D.P. Mishev In this case σ = 2 p. (4.3) Example 4..3: Let Y has geometric distribution, i.e., y σ y = PY ( = y) = p( p), < p<, y=,2,. (4.32) Then, probability generating function for this distribution is with pz S( z) =, (4.33) ( pz ) EY ( ) = σ = S () =. (4.34) p In this case, (4.9) becomes ( rμ ν / p)[ ( p) z]( z) qz ( ) =. 2 2 rμ( z) + rμp( z) z ν pz (4.35) Hence, from (4.35) we have L SS ρ(2 ρ) =. ρ (4.36) 4.2 Algorithm for recursively solving the system (4.5) Thus, using pgf, we found mean. We can also easily find variance, if necessary. However, conversion of the pgf to obtain distribution may be a difficult task, depending upon the type of S(z). Hence, we solve the system (4.5) for q n recursively. But, since a closed form solution seems formidable, we offer an algorithmic approximate solution as follows: Step 4. Choose data. Step 4.2 Choose ν, μ and r. For the purpose of computation, choose a fixed number m for maximum value of number of terms in probability distributions of number of applications in the SS. Set M = mr. Initial value of M should be such that all terms of σ Y after M are zero. Choose σ Y, a distribution function for Y. One may choose different values for the parameter(s) of distribution of σ Y. Write σ n. Revise M such that probabilities of number of phases in SS will be after the M th term. This value will be revised again later to make sure that sum of probabilities of number of applications in SS is. Step 4.3 Calculate σ.

25 Stochastic three-stage hiring model as a tandem queueing process 595 Step 4.4 ν Compute ρ =. rμ Step 4.5 Compute value of q = ρσ, given in (4.2). Step 4.6 Step 4.7 Adjust ν and μ for the following stability condition to be satisfied: < ρ <. (4.37) σ The reason for (4.37) is because, q is a probability and must be between and and for the sake of avoiding degeneration, we exclude and. Thus, (4.34) follows from Step 4.6. Recursively, find b n+, where q = ρ =. (4.38) q rμ b ν and b q n = (4.39) n+ n+,,2,, qn Note that q q q qn = = q q q q bb b n 2 n 2 n q q q q q q q q b b b 2 n and = =, etc. n 2 3 n 2 3 n Thus, write the second equation of (4.5) as: (4.4) Step 4.8 σ σ σ bn + = ρ + ρ n = M 2 n,, 2,,. bn bnbn bnbn b (4.4) Note that in (4.4), terms in bracket are chosen according to the value of n in (4.5). For instance, for n =, only the first term will be counted, etc. Calculate probability of the number of phases, q n. Here is how: Note that from Step 4.6 and (4.39), we have q = bq, q = b q = bb q,, q = bb b q. (4.42) n+ 2 n The set of relations in (4.42) is the distribution of the number of phases in SS that we are looking for. From (4.42) and the normalising equation below (4.5), we will have q (4.43) n = + b. n= i= i

26 596 A.M. Haghighi and D.P. Mishev Step 4.9 Thus, using M in Step (or its revisions) as. Find the sum of products in bracket in (4.43). Then, q from (4.43) and q n from (4.42). Make sure the sum is, otherwise revise parameters chosen above until all probabilities, q n, after the M th term are. Check the difference between q found from (4.42) and the one computed in Step from (4.2). This should give an error analysis for taking M in Step (or its revisions) as. Step 4. To compute Q n, from (4.24), let Q = q = ρσ. Then, use q from Step 8, using (4.24), compute Q n. Check to make sure the sum is. For the sake of example of how the algorithm works, we choose two distributions for Y, namely, Bernoulli with probability of success (defined below) as p and geometric with parameter p. We leave the table of probabilities in the Appendix. All rows with all elements have been truncated except for one. This is to show for what value of M the has been reached. Example 4. Bernoulli: We choose σ = p and σ 2 = p. In other words, success in this case means the batch size of and failure means a batch size of 2. Then, σ = 2 p. Now for ν =, μ =, and r = 5, we will have ρ =.2. Then, from this data, for values of p as.,.2, /3,.5,.6, 2/3, and 7/8, m = 5 and M 25, Tables 3 to 5 contain the amount of errors, Q n, and q n, respectively. Figure 2 shows the Bernoulli probabilities and cumulative probabilities of number of tasks and phases in the SS. Table 3 Amount of errors for numerical example (A), Bernoulli distribution p q = ρσ q = + n b i n= i= % Difference between approximated q and q = ρσ Note: *.e-5 Table 4.742*.2287*.386*.26*.4*.*.* Probability of number of applications for interview, Q n, in SS for numerical example (A), Bernoulli distribution p