An ASIP model with general gate opening intervals

Queueing Systems, Jul 2016

We consider an asymmetric inclusion process, which can also be viewed as a model of n queues in series. Each queue has a gate behind it, which can be seen as a server. When a gate opens, all customers in the corresponding queue instantaneously move to the next queue and form a cluster with the customers there. When the nth gate opens, all customers in the nth site leave the system. For the case where the gate openings are determined by a Markov renewal process, and for a quite general arrival process of customers at the various queues during intervals between successive gate openings, we obtain the following results: (i) steady-state distribution of the total number of customers in the first k queues, \(k=1,\dots ,n\); (ii) steady-state joint queue length distributions for the two-queue case. In addition to the case that the numbers of arrivals in successive gate opening intervals are independent, we also obtain explicit results for a two-queue model with renewal arrivals.

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An ASIP model with general gate opening intervals

An ASIP model with general gate opening intervals Onno Boxma 0 1 2 Offer Kella 0 1 2 Uri Yechiali 0 1 2 B Onno Boxma 0 1 2 Uri Yechiali 0 1 2 0 Department of Statistics and Operations Research, School of Mathematical Sciences, Tel Aviv University , 69978 Tel Aviv , Israel 1 Department of Statistics, The Hebrew University of Jerusalem , Mount Scopus, 91905 Jerusalem , Israel 2 EURANDOM and Department of Mathematics and Computer Science, Eindhoven University of Technology , P.O. Box 513, 5600 MB Eindhoven , The Netherlands We consider an asymmetric inclusion process, which can also be viewed as a model of n queues in series. Each queue has a gate behind it, which can be seen as a server. When a gate opens, all customers in the corresponding queue instantaneously move to the next queue and form a cluster with the customers there. When the nth gate opens, all customers in the nth site leave the system. For the case where the gate openings are determined by a Markov renewal process, and for a quite general arrival process of customers at the various queues during intervals between successive gate openings, we obtain the following results: (i) steady-state distribution of the total number of customers in the first k queues, k = 1, . . . , n; (ii) steady-state joint queue length distributions for the two-queue case. In addition to the case that the numbers of Onno Boxma: research done in the framework of the IAP BESTCOM project, funded by the Belgian government and by the Gravity program NETWORKS of the Dutch government. Offer Kella: supported in part by Grant 1462/13 from the Israel Science Foundation and the Vigevani Chair in Statistics. - arrivals in successive gate opening intervals are independent, we also obtain explicit results for a two-queue model with renewal arrivals. Mathematics Subject Classification 60K25 · 90B22 1 Introduction The asymmetric inclusion process (ASIP), introduced and analyzed in [ 5–9 ], is a one-dimensional lattice of n sites (queues), where particles (for example, customers) arrive randomly into the first site (Q1), stay there (‘served’) for a random time, continue moving simultaneously and unidirectionally from site to site while staying for a random time in each site, until finally exiting the last site (Qn ) and leaving the system. The ASIP defines the missing link between the celebrated Tandem Jackson Network (TJN) and the Asymmetric Exclusion Process (ASEP) [ 1–3 ] which plays the role of a paradigm in nonequilibrium statistical mechanics. Imagine that each site has a gate behind it that opens every exponentially distributed random time, allowing particles in the site to move forward to the next site. Denoting by Ccapacity the capacity of a site (i.e., maximal number of particles that can reside in the site), and by Cgate the capacity of the site’s gate (i.e., the maximal number of particles that can move forward when the gate opens), then, for the TJN, Ccapacity = ∞ and Cgate = 1, while for the ASEP Ccapacity = Cgate = 1. When Ccapacity = Cgate = ∞, one obtains the ASIP model where, when a gate of a site opens, all particles (customers) present there move simultaneously to the next site, joining the particles already there and forming a cluster of particles that continues moving as one unit. In the present work, we generalize the ASIP model by assuming that gate openings are determined by a Markov renewal process such that if, at some time, gate i opens, then with probability pi j the next gate to open is gate j , and the time until that gate opens is a random variable Oi j . We derive the Probability Generating Function (PGF) of the total occupancy (i.e., total number of customers) of sites 1 to k (k = 1, 2, . . . , n), while further studying the case when pi j = q j . We obtain the joint queue length distribution for the two-queue case, and analyze the system assuming binomial movement of particles. That is, when gate i (say) opens, each particle from the Xi particles present in site i (Qi ) will move forward (independently of the other particles) to site i + 1 (Qi+1) with probability ai , such that the total number of particles moving from Qi to Qi+1 is binomially distributed with parameters Xi and ai . The ASIP model, first studied in [ 6 ], presents a one-dimensional lattice of n queues with Poissonian flow only into the first site. Each gate opens, distinctly and independently of the others, every exponential time with rate μk for site k, implying that Oi j is the same for all (i, j ) combinations, and exponentially distributed with rate μk , while pi j = μ j / μk , i, j = 1, 2, . . . , n. The multidimensional PGF of the occupancy vector (X1, . . . , Xn) was studied, and it was shown that this PGF does not exhibit the famous product-form solution characterizing Jackson Networks. Accordingly, an iterative solution procedure was developed. However, the PGF of the total-load-upto-site k, k = 1, 2, . . . , n, was shown to have a product-form solution of geometric variables. For various objective functions, it was shown that the optimal intensities of the gate openings should be equal to each other. Considering large-size ASIP, it was observed in [ 8 ], via simulations, that P ( Xk > 0) ∼ k−1/2, E [ Xk | Xk > 0] ∼ k1/2; and that (standard deviation of Xk )/E [ Xk ] ∼ k1/4. Those observations were later proved analytically in [ 7 ], where limit laws when n → ∞ were derived. Various measures were investigated: (i) a particle’s (customer’s) traversal time, T , in a homogeneous ASIP, is distributed as T ∼ nm + n1/2m Z , where m = mean time between successive gate openings, m2 is its variance, and Z is the Gaussian (0, 1) random variable. (ii) The Laplace–Stieltjes Transform (LST) and mean of the busy period (the time from the first arrival of a customer at an empty system until the first moment thereafter that the network becomes empty again). (iii) The LST and mean of the draining time (the time from an arbitrary moment when the system is in steady state and the inflow is stopped, until the first moment thereafter that the system becomes empty). Occupation probabilities were considered in [ 9 ]. Closed-form results were obtained for the probabilities that the total occupation of ‘lattice intervals’ of m sites, sites k to k + m − 1, is equal to l, l = 0, 1, 2, . . . . In particular, when l = 0, the problem becomes a discrete boundary value problem and the probabilities are derived with the aid of Catalan numbers. The main contribution of this paper is that it considerably extends the exact analysis of ASIP tandem models: We allow the gate openings to be determined by a Markov renewal process, instead of assuming that each gate opens after exponentially distributed intervals, and we extend the Poisson arrival assumption by allowing a quite general arrival process of customers at the various queues during intervals between successive gate openings. Under these assumptions, we determine the steady-state distribution of the total number of customers in the first k queues, k = 1, . . . , n. We obtain some additional results for the two-queue case, and solve three optimization problems, thus obtaining insight into the design of ASIPs. The paper is organized as follows. Section 2 contains the model description. Section 3 is devoted to the analysis of the steady-state joint distribution of the numbers of customers in the various queues just after gate openings. Section 4 contains a brief discussion on optimization of the system performance, for the case that arrivals only occur in Q1. A few more detailed two-queue results are presented in Sect. 5. We conclude with some suggestions for further research in Sect. 6. 2 Model description Consider the following model of n queues Q1, . . . , Qn in series. Each queue has one gate behind it, which may be viewed as a server. Gates are almost all the time closed. When gate i (the gate behind Qi ) opens, all customers present in Qi are instantaneously transferred to Qi+1, i = 1, 2, . . . , n − 1; when gate n opens, all customers present in Qn instantaneously leave the system. After the transfer, the gate immediately closes again. Gate openings are determined by a Markov renewal process. If, at some time t , gate i opens, then with probability pi j the next gate to open is gate j ; and the time until that gate opens is a random variable Oi j . We assume that the Markov chain governing the successive gate openings is irreducible and we denote its steady-state distribution by πi , i = 1, . . . , n. During an Oi j period, customers may arrive at all queues. We assume that the vectors of arrival numbers in successive gate opening intervals are independent, but may depend on the indices i and j . The generating function of the numbers of arrivals into Q1, . . . , Qn during an Oi j period is given by Ai j (z1, . . . , zn ). In addition, we denote the generating function of the cumulative number of arrivals into Q1, . . . , Qk during an Oi j period by Ai jk (z) := Ai j (z, . . . , z, 1, . . . , 1), where the last z occurs at position k. Notice that one example is provided by a batch Poisson arrival process, possibly with dependence between batch sizes at different queues, and with arrival rates which may depend on the type of gate opening interval. The ASIP model as introduced and studied in [5–9] is a generic model that may represent many different stochastic processes in chemistry, physics, and everyday life. From a queueing perspective, it is a series of queues with unlimited batch service. The notion of batch service is closely related to growth-collapse processes. Stochastic growth-collapse temporal patterns appear in a variety of systems, like sandpile models and systems in self-organized criticality, stick-slip models of interfacial friction, Burridge–Knopoff models of earthquakes and continental drift, stochastic avalanche models, and stochastic Ornstein–Uhlenbeck capacitors (cf. page 16 of [ 5 ], and references given there). From a statistical physics perspective, the ASIP is a reaction–diffusion model for unidirectional transport with coagulation. Our model significantly generalizes the model of [ 5–9 ], allowing us to more accurately represent those stochastic processes. It also allows one to represent movements of ships, crowds, or cars. An ASIP model may represent a series of sluices, with ships simultaneously moving from one section to the next one when a gate is opened. An ASIP may also represent the movement of a crowd through a series of sections of an amusement park—and in both settings it is more natural to model the gate openings by a Markov renewal process than by assuming that all gates open according to independent Poisson processes (independent exponential gate opening intervals). Furthermore, in several settings, for example in a series of road traffic intersec tions with traffic lights, it is also restrictive to only allow external arrivals at Q1. In our model, during a gate opening interval Oi j , arrivals at all the queues are possi ble. We do restrict ourselves to the case in which customers from Qi can only move to Qi+1, i = 1, 2, . . . , n − 1. That assumption will allow us to obtain exact results for the total number of customers X(k) which are present in the first k queues right after a gate opening (k = 1, 2, . . . , n). Our results will become somewhat simpler in the special case in which the next gate opening is of gate j with a fixed probability q j , i.e., irrespective of the index of the previous gate opening. Notice that the original ASIP model of [ 5–9 ] also has this property, as there the gate openings are governed by independent Poisson processes. We work out this special case of fixed gate opening probabilities q j in an example in Sect. 3, showing that just like in [ 6,7 ] X(k) can be written as the sum of k independent random variables. We also consider the problem of optimally choosing the gate opening fractions q j in Sect. 4. 3 Analysis We are interested in the steady-state joint distribution of the numbers of customers (X1, . . . , Xn) just after a gate opening. To argue the existence of such a distribution, let ξk = (ξk1, . . . , ξkn) be the state of the network right after the kth gate opening and let gk be the gate that opened. Then, because the external arrival process is independent of the process of the gate openings, (ξk , gk ) is a Markov chain. To argue that it is positive recurrent (on an appropriate state space), let us define an auxiliary process as follows. Let ηki = I (ξki ≥ 1). Then (ηk , gk ) is also a Markov chain and ξk is the zero vector if and only if ηk is. We note that for every station j , the state (0, j ) is accessible from every other state. This is because if we block external arrivals, the time until the network becomes empty is finite (actually has a finite expectation). When this happens, we are in some state (0, ). Since gk is irreducible, then, if we once again block all arrivals the state (0, j ) is accessible from (0, ) (actually, without arrivals, it will also be reached after finite expected time). With positive probability, the time until the first arrival is greater than the (independent) time to reach (0, j ) without arrivals and thus (0, j ) is accessible from any other state. Thus, on the states (y, j ) which are accessible from (0, j ) (which include (0, ) for all 1 ≤ ≤ n and all states that are accessible from (0, ) for any such ), we have that (ηk , gk ) is an irreducible Markov chain and since the state space is finite (contained in or equal to {0, 1}n × {1, . . . , n}) it follows that it is positive recurrent. Therefore, for any j , the time between visits to state (0, j ) has a finite mean. This implies that the time between visits of (ξk , gk ) to (0, j ) also has a finite mean and thus the (ξk , gk ) is also positive recurrent on an appropriate state space (all the states which are accessible from (0, j ) for some, hence all, j ). We note that this idea can be used to argue stability for the continuous time process, which although it is not semi-Markov due to the arrival process, is nevertheless regenerative with finite mean regeneration epochs, provided that Oi j have finite means. Since we do not need it here, we omit the details. In the present section, we shall in particular focus on X(k) := X1 + · · · + Xk , viz., the total number of customers in the first k queues right after a gate opening. Introducing M , the index of the gate that has just opened, we consider Gki (z) := E[z X(k) I (M = i )], k, i = 1, . . . , n, (1) where I (·) denotes an indicator function. The fact that customers can only move to downstream queues (i.e., with higher index) will allow us to express all Gki (z) for a fixed k in terms of functions Gk−1, j (z), and finally in terms of the functions G1 j (z), which can be determined explicitly. We begin by giving the equations for G1 j (z), j = 1, . . . , n. Obviously G11(z) = P(M = 1) = π1; (2) indeed, after gate 1 has opened, Q1 instantaneously has become empty. Now consider two successive gate openings in steady state, the latter one being an opening of gate j , and sum over all possible gates i opened at the previous gate opening, to obtain Here, we have used that Ai j1(z) is the generating function of the number of arrivals at Q1 in the gate opening interval. Notice that we can rewrite (3) as (3) (5) (6) (7) (8) n i=2 G1 j (z) = G1i (z) pi j Ai j1(z) + G11(z) p1 j A1 j1(z), j = 1. (4) Introducing the (n − 1)-dimensional vector the (n − 1)-dimensional vector G¯ 1(z) := (G12(z), . . . , G1n(z)), R1(z) := ( p12 A121(z), . . . , p1n A1n1(z)), and the matrix P1(z) with as (i, j ) element pi j Ai j1(z), we can write (4) as G¯ 1(z) = G¯ 1(z) P1(z) + G11(z)R1(z), and hence, with I the matrix with ones on the diagonal and zeroes outside the diagonal, we have G¯ 1(z)(I − P1(z)) = G11(z)R1(z), yielding G¯ 1(z) = G11(z)R1(z)(I − P1(z))−1. All the terms on the right-hand side of (6) are known; in particular, G11(z) = π1 is given in (2). Hence we have determined G11(z), G12(z), . . . , G1n(z). Now let us show how the terms Gk j (z), j = 1, . . . , n, are for 2 ≤ k ≤ n expressed in terms of Gk−1,i (z), i = 1, . . . , n. Considering two successive gate openings in steady state, the last one being of gate j , and summing over all possible gates i for the first gate opening, we have for k = 2, . . . , n, j = k: n i=1 n i=1 The explanation for the deviating terms (Gk−1,i (z) instead of Gki (z) and Aik,k−1(z) instead of Aikk (z)) is that Qk has become empty right after an opening of gate k; so the total number present in Q1, . . . , Qk equals the total number present in Q1, . . . , Qk−1 after the previous gate opening, plus the number of new arrivals in the first k − 1 queues. We can rewrite (7) as follows: Rk (z) := ( pk1 Ak1k (z), . . . , pk,k−1 Ak,k−1,k (z), pk,k+1 Ak,k+1,k (z), . . . , pkn Aknk (z)), and the matrix Pk (z) with as (i, j ) element pi j Ai jk (z), we can write (9) as G¯ k (z) = G¯ k (z) Pk (z) + Gkk (z)Rk (z), G¯ k (z) = Gkk (z)Rk (z)(I − Pk (z))−1. (10) (11) (12) (13) (14) We have thus expressed G¯ k (z) in terms of Gkk (z) via (11), and Gkk (z) in terms of G¯ k−1(z) and Gk−1,k−1(z) via (12). Iterating, defining an empty product to be one and defining G¯ 0(z)C0T (z) to equal π1 for notational elegance, we obtain By carefully studying the structure of the above recursions, and introducing k−1 i=0 k−1 j=i+1 Hi (z) := Ri (z)(I − Pi (z))−1CiT (z), i = 1, . . . , n, the following is seen to hold: k−1 i=1 Gkk (z) = π1 Fi (z), k = 1, . . . , n, where denotes a sum over the 2k−1 terms that arise when each Fi (z), i = 1, . . . , k−1, is either Hi (z) or pi,i+1 Ai,i+1,i (z). For example, for k = 3 we get G33(z) = π1[H1(z)H2(z) + H1(z) p23 A232(z) + p12 A121(z)H2(z) With this explicit expression (14) for the Gkk (z), and expression (11) for G¯ k (z), we have a recipe to determine all Gk j (z) explicitly, for k, j = 1, . . . , n. Example Let us consider the special case in which pi j ≡ q j , ∀ i, j , and Ai jk (z) =: Aˆ jk (z), ∀ i, j, k. Viz., the Markov renewal process that determines the gate openings and the intervals in between has a simple structure: Each time the next gate opening is of gate j with probability q j , and the interval length until the next opening also only depends on j . In this case, we can obtain a simple expression for E[z X(k) ] = n j=1 Gk j (z). We have G11(z) = π1 = q1, and from (3) Hence yielding Furthermore, from (7) and (8), n j=1 G1 j (z) = q j Aˆ j1(z) G1i (z), j = 2, . . . , n. E z X(1) = G1 j (z) = q1 + q j Aˆ j1(z)E[z X(1) ], n i=1 n j=2 q1 n i=1 n i=1 E[z X(1) ] = 1 − n j=2 q j Aˆ j1(z) . Gk j (z) = q j Aˆ jk (z) Gki (z), Gkk (z) = qk Aˆk,k−1(z) Gk−1,i (z), leading to the following recursive expression for E[z X(k) ] in terms of E[z X(k−1) ]: E[z X(k) ] = 1 − qk Aˆk,k−1(z) j =k q j Aˆ jk (z) E[z X(k−1) ]. (15) (16) (17) (18) (19) (20) (21) , where Aˆ10(z) := 1. Notice that (22) represents a decomposition property: The generating function is a product of k terms, all of which are generating functions of random variables, and this implies that X(k) can be represented as the sum of k independent random variables, cf. [ 6,7 ]. In the special case that arrivals only occur at Q1, and that the generating function of the number of arrivals in all gate intervals is the same, to be denoted by Aˆ(z), we have E[z X(k) ] = Aˆk−1(z) k qi i=1 1 − Aˆ(z)(1 − qi ) . When we consider for this case the steady-state number of customers N(n) just before a gate opening, we get a slightly more elegant expression. Observing that E[z N(n) ] = E[z X(n) ] Aˆ(z), we can write E[z N(n) ] = n qi Aˆ(z) i=1 1 − Aˆ(z)(1 − qi ) . This shows that N(n) is distributed like the sum of n independent geometric sums of numbers of arrivals during one gate interval. In particular, Via iteration we obtain (22) (23) (24) (25) (26) (27) EN(n) = E A Var N(n) = (E A)2 n 1 i=1 qi n , i=1 1 1 qi2 − qi + Var A n 1 i=1 qi . A denoting the number of arrivals during one gate interval. The special choice Aˆ(z) = z (one arrival in each gate interval) yields E[z X(k) ] = zk−1 k i=1 qi 1 − (1 − qi )z , and hence X(k) = k − 1 +1−qi ik==1 Bi , where Bi ∼ geom(1 − qi ) for i = 1, . . . , k, and EX(k) = k − 1 + ik=1 qi ik=1 q1i − 1. The special choice Aˆ(z) = μ+λμ(1−z) (a Poisson distributed number of arrivals in an exp(μ) distributed interval, giving rise to a geometrically distributed number of arrivals in a gate interval) yields (28) is geom( qiqμi μ+λ ) distributed with probability 1 − qi , i = 1, . . . , k. Hence EX(k) = (k − 1) μλ + ik=1(1 − qi ) qiλμ . More generally, it follows from (23) that EX(k) = 4 Optimization under constraints In this section we consider three optimization problems, which are very similar to optimization problems studied in [ 6 ] for the special case of exponential gate openings. Our goal is to design an efficient ASIP system. We restrict ourselves to the case, leading to (24), in which arrivals only occur in Q1 while the generating function of the number of arrivals in all gate intervals is the same. For this case, we pose the question which choice of (q1, . . . , qn ), with in=1 qi = 1, (i) minimizes the mean number of customers N(n) just before a gate opening, (ii) minimizes the variance of N(n), and (iii) maximizes the probability of zero load (an empty system). Optimization problem (i): minimization of the mean number of customers It follows from (25) that the minimization of the mean number of customers amounts to minimizing in=1 q1i , subject to in=1 qi = 1. This optimization problem is a special case of the class of resource allocation problems with a separable convex objective function i.e., the objective function can be separated into n terms, the i th one being a function of qi only, that is convex in qi , i = 1, . . . , n. We wish to minimize this separable convex function under a linear constraint. This class of problems is extensively studied in [ 4 ]. In particular, if f is convex then f (1/n) = f 1 n n i=1 qi 1 n ≤ n i=1 f (qi ) and so in=1 f (1/n) ≤ in=1 f (qi ) for any qi ≥ 0 such that in=1 qi = 1; thus the optimal solution of our minimization problem is q1 = · · · = qn = n1 . It should be noted that the mean number of customers just before a gate opening is readily expressed in the steady-state mean number of customers at an arbitrary epoch; just subtract the mean number of arrivals in a residual gate opening interval. The latter is linearly related to the mean time in system via Little’s law. Hence the above optimization problem also sheds light on the minimization of time in system. Optimization problem (ii): minimization of the variance of the number of customers It follows from (26) that the minimization of the variance of the number of customers amounts to minimizing in=1[( q1i2 − q1i )(E A)2 + q1i Var A]. The same reasoning as for (i) applies; we again are faced with a separable convex objective function, and again the optimal solution is q1 = · · · = qn = n1 . Optimization problem (iii): maximization of the probability of an empty system It follows from (24) that the maximization of the probability of an empty system amounts to maximizing in=1 1−(q1i−Aˆq(i0))Aˆ(0) , and hence to minimizing in=1 ln[ 1−qAˆi(0) + Aˆ(0)]. The same reasoning as for (i) and (ii) applies once again; we have a separable convex objective function, and again the optimal solution is q1 = · · · = qn = n1 . 5 Some two-queue results In this section we study the two-queue case in some more detail. In that case one can sometimes determine the joint queue length distribution at gate opening intervals. In Sect. 5.1 we determine the joint queue length distribution at gate openings for a specific choice of the pi j and the same arrival distributions for gate 1 intervals and gate 2 intervals. In Sect. 5.2 we determine the joint queue length distribution for the case in which, when the gate of Q1 opens, only a binomially distributed number of the customers in Q1 moves to Q2. These two-queue studies not only lead to more detailed results, they also sometimes give an indication of the limitations of our approach. For example, if one would not only at Q1, but also at Q2, allow a binomially distributed number of customers to leave when its gate opens, then a functional equation in the two-dimensional queue length probability generating function results, which seems very difficult to analyze exactly. 5.1 Joint queue length distribution Let us consider the problem of determining the generating function of the steady-state joint queue length distribution right after gate openings, G(z1, z2) = E z X1 z2X2 ]. Take [ 1 n = 2; take only arrivals at Q1, with generating function A(z) of the number of arrivals per gate opening, regardless whether it is an opening of gate 1 or of gate 2; and take fixed gate opening probabilities pi j ≡ q j . Realizing that, with Xi(r) the number of customers in Qi right after the r th gate opening, and with Ar+1 the number of arrivals in the interval between the r th and (r + 1)st gate openings, X 1(r+1) = 0, X 2(r+1) = X 1(r) + Ar+1 + X 2(r), if the (r + 1)st gate opening is of gate 1, and X 1(r+1) = X 1(r) + Ar+1, X 2(r+1) = 0, if the (r + 1)st gate opening is of gate 2, we obtain in steady state: G(z1, z2) = q1 A(z2)G(z2, z2) + q2 A(z1)G(z1, 1). (29) Actually we already know G(z1, 1), which equals E[z1X(1) ]; but it also follows from (29) by putting z2 = 1. We also already know G(z1, z1), which equals E[z1X(2) ]; but it also follows from (29) by putting z2 = z1. We find (30) (31) (32) Remark One could extend the above analysis to the case of arrivals at both queues, and different PGFs for different gate openings. However, this comes at the expense of messier expressions, and we have decided not to include this case in the paper. One could also in principle analyze the steady-state queue length PGF at an arbitrary epoch. One would then have to average over different gate opening intervals. However, the arrival process must then first be specified in more detail; do arrivals all take place at the beginning of a gate opening interval, or at the end, or maybe according to a Poisson process? 5.2 Binomial movements Consider the case of n = 2 queues in series, with the special feature that, when the gate of Q1 opens, each customer present in Q1 (independently from the other customers) moves with probability a1 > 0 to Q2, and stays with probability 1 − a1 in Q1. We restrict ourselves to the case of a Poisson arrival process, with rate λ, at Q1, and no external arrivals at Q2; moreover, we assume that gate openings at Qi occur after i.i.d., exponentially distributed intervals with mean 1/μi , i = 1, 2. Denoting by Xi (t ) the number of customers in Qi at time t , i = 1, 2, and by X bin(t ) the number of customers 1 who do move from Q1 to Q2 at a gate opening of Q1 that takes place at time t , we can write (suppressing initial conditions; we shall anyway soon turn to the steady-state situation) E z X1(t+h)z X2(t+h) 1 2 = (1 − (λ + μ1 + μ2)h)E z X1(t)z X2(t) 1 2 + λhz1E z X1(t)z X2(t) 1 2 + μ1hE z X1(t)−X1bin(t)z X2(t)+X1bin(t) 1 2 + μ2hE z X1(t) 1 + o(h), h ↓ 0, (33) d E z X1(t)z X2(t) dt 1 2 = − (λ + μ1 + μ2)E z X1(t)z X2(t) 1 2 + λz1E z X1(t)z X2(t) 1 2 leading to (34) Denoting the probability generating function of the joint distribution of the steady-state queue length vector (X1, X2) by H (z1, z2), we have [μ1 + μ2 + λ(1 − z1)]H (z1, z2) = μ1 H ((1 − a1)z1 + a1z2, z2) + μ2 H (z1, 1). (35) We shall first obtain H (z1, 1). Substituting z2 = 1 into (35) yields [μ1 + μ2 + λ(1 − z1)]H (z1, 1) = μ1 H ((1 − a1)z1 + a1, 1) + μ2 H (z1, 1), (36) and hence Iteration of this relation gives μ1 H (z1, 1) = μ1 + λ(1 − z1) H ((1 − a1)z1 + a1, 1). H (z1, 1) = ∞ j=0 μ1 μ1 + λ(1 − z1)(1 − a1) j . This infinite product is said to converge iff ∞j=0 1 − μ1+λ(1−μz11)(1−a1) j < ∞, and hence the infinite product indeed converges if 0 < a1 < 1. If a1 = 1 one obtains μ1 H (z1, 1) = μ1+λ(1−z1) . This is not a surprising result; it is the generating function of the number of Poisson(λ) arrivals during an exp(μ) interval. According to PASTA, it also equals the generating function of the steady-state queue length distribution of Q1. Observing that μ1+λ(1−μz11)(1−a1) j is the probability generating function of a geometrically distributed random variable with success parameter μ1λ+(1λ−(1a−1a)j1) j , one can write d ∞ X1 = H j , j=0 (37) (38) (39) where all H j are independent, H j being geometrically distributed with success parameter μ1λ+(1λ−(1a−1a)j1) j . Having determined H (z1, 1), we now turn to the determination of H (z1, z2). It follows from (35) that H (z1, z2) = Y1(z1)H ((1 − a1)z1 + a1z2, z2) + Y0(z1), (40) where Y1(z1) := μ1 + μ2 +μ1λ(1 − z1) , Y0(z1) := μ1 + μ2 +μ2λ(1 − z1) H (z1, 1). Iteration of (40) gives H (z1, z2) = Y0( f j (z1, z2)) Y1( fi (z1, z2)), an empty product being equal to one and fi (z1, z2) := (1 − a1)i z1 + [1 − (1 − a1)i ]z2, i = 0, 1, . . . . Using d’Alembert’s ratio test one can show that this infinite sum converges. In fact, the sum converges geometrically fast. Indeed, since a1 > 0, one has f j (z1, z2) → z2, and the ratio of two successive terms in the sum H (z1, z2), which is given by YY0(0(f jf+j (1z(1z,1z,2z)2))) Y1( f j (z1, z2)), is for large j bounded by μ1/(μ1 +μ2). Above, we have restricted ourselves to the case of a Poisson arrival process, with rate λ, at Q1, and no external arrivals at Q2; moreover, we assumed that gate openings at Qi occur after i.i.d. exponentially distributed intervals with mean 1/μi , i = 1, 2. Let us now turn to the more general case of Sect. 2, in which gate openings are determined by a Markov renewal process, and where a gate opening of Qi is with probability pi j followed by a gate opening of Q j , while Ai j (z1, z2) is the generating function of the numbers of arrivals in Q1 and Q2 during the period in between those two successive gate openings. Considering the steady-state joint distribution of the numbers of customers (X1, X2) immediately after gate openings, and letting (cf. (2)) Gi (z1, z2) := E[z1X1 z2X2 I (M = i )], i = 1, 2, (42) it is easily seen by observing the system at two successive gate openings that G1(z1, z2) = p11 A11((1 − a1)z1 + a1z2, z2)G1((1 − a1)z1 + a1z2, z2) G2(z1, z2) = p12 A12(z1, 1)G1(z1, 1) + p22 A22(z1, 1)G2(z1, 1). + p21 A21((1 − a1)z1 + a1z2, z2)G2((1 − a1)z1 + a1z2, z2), (43) (44) It is immediately obvious from (44) that G2(z1, z2) does not depend on z2, as we could have expected, because Q2 becomes empty after a gate opening at Q2. Hence, it follows from (44) that Plugging z2 = 1 in (43) and using (45) gives G1(z1, 1) = p11 A11((1 − a1)z1 + a1, 1)G1((1 − a1)z1 + a1, 1) p12 A12((1 − a1)z1 + a1, 1) + p21 A21((1 − a1)z1 + a1, 1) G1((1 − a1)z1 + a1, 1), 1 − p22 A22((1 − a1)z1 + a1, 1) (41) (45) (46) which can be written as G1(z1, 1) = L(z1)G1((1 − a1)z1 + a1, 1), with an obvious choice of the function L(·). Iteration readily yields that where where G1(z1, 1) = L(d( j)(z1)), ∞ The infinite product converges iff the corresponding infinite sum ∞j=0[1 − L(d( j)(z1))] converges. The latter sum converges geometrically fast. This can be seen by making the following two observations. Observation (i): L(z1) has the meaning of a probability generating function. Indeed, distinguish between the possibility that a gate opening of Q1 is followed by another gate opening of Q1 (probability p11) and the possibility that it is followed by a gate opening of Q2, followed by a geometric( p22) number of gate openings of Q2, and finally again a gate opening of Q1. Observation (ii): 1 − d( j)(z1) = (1 − a1) j (1 − z1) converges geometrically fast to 0. Having determined G1(z1, 1) and hence, using (45), G2(z1, z2) = G2(z1, 1), we substitute the result in (43), obtaining G1(z1, z2) = K1(z1, z2)G1((1 − a1)z1 + a1z2, z2) + K0(z1, z2), G1((1 − a1)z1 + a1z2, 1). K1(z1, z2) := p11 A11((1 − a1)z1 + a1z2, z2), Again d’Alembert’s ratio test readily shows the convergence of the infinite sum, by using that |K1(z1, z2)| < p11. Finally, notice that G1(z2, z2), which is the generating function of the total number of customers X(2) = X1 + X2 in the two queues just after gate openings of Q1, follows by substituting z1 = z2 in (53). Since f j (z2, z2) ≡ z2, that formula degenerates into (47) (48) (49) (50) (51) (52) (53) (54) After some calculations, this expression is seen to agree with the expression for G21(z2) that can be derived from (7). This agreement may at first sight seem strange, as we have binomial movements in the present subsection. However, notice that we compare G1(z2, z2) and G21(z2), both giving the total number of customers in both queues. It then does not matter whether some of them are still in Q1 after a gate opening of Q1. 5.3 An ASIP model with a renewal arrival process at Q1 In this subsection we consider the case in which arrivals only take place at Q1, and follow a renewal process: successive interarrival times are i.i.d., with distribution A(·) and Laplace–Stieltjes transform α(·). We restrict ourselves to n = 2 queues. We furthermore restrict ourselves to the case in which openings of the gate of Qi occur at i.i.d. exp(μi ) distributed intervals, independent of each other and independent of the arrival intervals. Let (Yn,1, Yn,2) denote the vector of numbers of customers in (Q1, Q2) just before the nth arrival at Q1, n = 1, 2, . . . . Let An denote the arrival interval between customers n − 1 and n. We need to distinguish between the following five cases: (i) No gate opening in An. This event has probability α(μ1 +μ2); and (Yn,1, Yn,2) = (Yn−1,1 + 1, Yn−1,2). (ii) No openings of gate 1 and at least one opening of gate 2 in An. This event has probability α(μ1) − α(μ1 + μ2); and (Yn,1, Yn,2) = (Yn−1,1 + 1, 0). (iii) No openings of gate 2 and at least one opening of gate 1 in An. This event has probability α(μ2) − α(μ1 + μ2); and (Yn,1, Yn,2) = (0, Yn−1,1 + 1 + Yn−1,2). (iv) Both gates open at least once in An; the first opening of gate 1 occurs after the μ2 last opening of gate 2. This event has probability α(μ1 + μ2) − μ2−μ1 α(μ2) − μ1 μ1−μ2 α(μ1); and (Yn,1, Yn,2) = (0, Yn−1,1 + 1). (v) Both gates open at least once in An ; but the first opening of gate 1 occurs before the μ2 μ1 last opening of gate 2. This event has probability 1− μ2−μ1 α(μ1)− μ1−μ2 α(μ2), as can, for example, be seen by writing the probability of this event as the probability that the sum of an exp(μ1) plus an exp(μ2) random variable is less than An. We now have (Yn,1, Yn,2) = (0, 0); notice that this is the only way to get into the state (0, 0). Restricting ourselves to steady-state queue lengths just before arrivals, to be denoted by (Y1, Y2), and introducing their generating function L(z1, z2) := E[z1Y1 z2Y2 ], we obtain L(z1, z2) = α(μ1 + μ2)z1 L(z1, z2) + (α(μ1) − α(μ1 + μ2))z1 L(z1, 1) + (α(μ2) − α(μ1 + μ2))z2 L(z2, z2) + 1 − μ2 μ−2 μ1 α(μ1) − μ1 μ−1 μ2 α(μ2). + α(μ1 + μ2) − μ2 μ−2 μ1 α(μ2) − μ1 μ−1 μ2 α(μ1) z2 L(z2, 1) (55) Taking all L(z1, z2) terms together, and introducing (which actually is L(0, 0) = P(Y1 = 0, Y2 = 0); see above) and ζ := 1 − μ2 μ−2 μ1 α(μ1) − μ1 μ−1 μ2 α(μ2), ω := α(μ1 + μ2) − μ2 μ−2 μ1 α(μ2) − μ1 μ−1 μ2 α(μ1) = α(μ1 + μ2) − α(μ1) − α(μ2) + 1 − ζ, we obtain and hence (1 − α(μ1 + μ2)z1)L(z1, z2) = (α(μ1) − α(μ1 + μ2))z1 L(z1, 1) + (α(μ2) − α(μ1 + μ2))z2 L(z2, z2) + ωz2 L(z2, 1) + ζ. Substitution of z2 = 1 in (56), and using the fact that α(μ2) − α(μ1 + μ2) + ζ + ω = 1 − α(μ1) yields (1 − α(μ1 + μ2)z1)L(z1, 1) = (α(μ1) − α(μ1 + μ2))z1 L(z1, 1) + 1 − α(μ1), (57) L(z1, 1) = E zY1 1 . The marginal distribution of Y1 is hence geometric. The explanation is that Y1 increases by 1 for a geometrically distributed number of arrival intervals (with parameter α(μ1), which is the probability that gate 1 does not close during an arrival interval), and then falls back to zero. Substituting z1 = z2 in (56) allows us to express L(z2, z2) in terms of L(z2, 1): (1−α(μ2)z2)L(z2, z2) = (α(μ1)−α(μ1 +μ2))z2 L(z2, 1)+ζ +ωz2 L(z2, 1), (59) yielding the following expression for the generating function of the total number of customers in the system just before an arrival at Q1: L(z2, z2) = (α(μ1) − α(μ1 + μ2) + ω) (11−−αα((μμ11)))zz22 + ζ Finally, Eqs. (56), (58), and (60) give the generating function of (Y1, Y2): 1 L(z1, z2) = 1 − α(μ1 + μ2)z1 + (α(μ2) − α(μ1 + μ2)) × + ω (61) (62) (63) (64) (65) Substituting z2 = 0 in (61) gives L(z1, 0) = (α(μ1) − α(μ1 + μ2)) (11−−αα((μμ11)))zz11 + ζ 1 − α(μ1 + μ2)z1 In a similar way we get L(0, z2) and L(1, z2) = E [z2Y2 ]. In particular, 1 L(1, z2) = 1 − α(μ1 + μ2) × (α(μ1) − α(μ1 + μ2)) + (α(μ2) − α(μ1 + μ2)) z2 (α(μ1) − α(μ1 + μ2)) (11−−αα((μμ11)))zz11 + ζ 1 − α(μ1 + μ2)z1 It is seen that the marginal distribution of Y2 has an atom in 0 and furthermore is a weighted sum of (i) a geometric(α(μ1)) distribution, (ii) a geometric(α(μ2)) distribution, and (iii) a convolution of two such geometric distributions. Finally, we determine the generating function of the joint distribution of the steadystate numbers of customers (S1, S2) in Q1 and Q2 at an arbitrary epoch. It is easily seen that this distribution is obtained by considering the queue lengths at a time Ar after the last customer arrival, where this forward recurrence interarrival time or residual interarrival time has LST αr (s) = 1−α(s) . We can follow the reasoning leading to (55), simply replacing each α(·) termsbEy[Aα]r (·). Hence = αr (μ1 + μ2)z1 L(z1, z2) + (αr (μ1) − αr (μ1 + μ2))z1 L(z1, 1) + (αr (μ2) − αr (μ1 + μ2))z2 L(z2, z2) + ω˜ z2 L(z2, 1) + ζ˜ , where ω˜ and ζ˜ are obtained from ω and ζ by replacing α(·) by αr (·) everywhere. 6 Suggestions for further research The following extensions might be of interest: 1. Firstly, and perhaps most interestingly, there are various asymptotic questions. For example, one could let n → ∞, and study, for example, the fraction of empty stations. We refer to Chapter 6 of [ 5 ] and to [ 7–9 ] for an interesting collection of limit laws for three limiting regimes (for the case of only arrivals at Q1, and exponential gate openings): (i) The heavy-traffic regime, in which the arrival rate at Q1 goes to infinity; (ii) the large-system regime in which n → ∞; (iii) the balanced-system regime, in which n → ∞, the gate opening intervals tend to zero, and the product of n and the mean gate opening interval tends to a positive limit. 2. We are presently exploring ASIP models with finite waiting rooms. In such a case it is, for example, interesting to allocate the waiting room sizes—under a constraint on total waiting room size—such that the throughput of the ASIP is as large as possible. 3. A batch can move one or two queues ahead at a gate opening. The approach taken in Sect. 3 to obtain expressions for the Gki (z) (cf. (1)) breaks down when batches could move more than one queue ahead after a gate opening. 4. At each gate opening, multiple gates can open. If, with probability ri , gates i and i + 1 open, i = 1, 2, . . . , then this amounts to a batch moving two queues ahead. So this variant is related to the previous one. 5. Nontandem configurations. For example, there are three queues, Q1 feeding into Q2 and Q3—with fixed probabilities, or via a fixed alternating pattern. Acknowledgments Onno Boxma gratefully acknowledges interesting discussions with Corné Suijkerbuijk. 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Onno Boxma, Offer Kella, Uri Yechiali. An ASIP model with general gate opening intervals, Queueing Systems, 2016, 1-20, DOI: 10.1007/s11134-016-9492-z