First exit time for a discrete-time parallel queue

Queueing Systems Vol. 100 Special Issue: 100 views on queues First exit time for a discrete time parallel queue arXiv:2202.08343v1 [math.PR] 16 Feb 2022 Zbigniew Palmowski February 18, 2022 1 Introduction We consider a discrete time parallel queue, which is a two-queue network, with batch arrivals and services. At the beginning of the nth time-slot, An customers arrive to both queues. Then after this, Sni ∈ N ∪ {0} customers can be served in slot n by server i for i = 1, 2. We assume that {An }{n∈N∪{0}}, {Sni }{n∈N∪{0}} are independent sequences of i.i.d. random variables with support in N ∪ {0} and EAn < ESni for i = 1, 2. Let i Qin with i = 1, 2 be the queue length after the service Sn−1 and before the arrival i i An . Then it satisfies the Lindley recursion Qn+1 = (Qn + An − Sni )+ for i = 1, 2 and its stationary law (Q1∞ , Q2∞ ) is given by (maxn∈N∪{0} Tn1 , maxn∈N∪{0} Tn2 ) where Tni = ∑nk=1 (Ak − Ski ), T0i = 0, i = 1, 2. For n An = ∑ Ak , k=1 Sni = n ∑ Ski , i = 1, 2, k=1 and x, y ∈ N ∪ {0}, we are interested in H(x, y) = P(Q1∞ > x, Q2∞ > y) = P(there exists n ∈ N such that An > max{x + Sn1 , y + Sn2 }). Note that H(x, y) equals the probability that the first entrance time of the two-dimensional random walk {(An − Sn1 , An − Sn2 ), n ∈ N∪{0}} to the set [x, ∞]× [y, ∞] is finite. In the context of actuarial science, H(x, y) corresponds to ruin probability for the a discretetime two-dimensional insurance risk model where each business line faces common simultaneous losses Ak and Ski , for i = 1, 2, is the premium derived within the kth period of time (see [1] for similar considerations). This two-queue network is a very special case of a discrete time network, but we hope that it provides very interesting insight in understanding the more general picture Department of Applied Mathematics, Wroclaw University of Science and Technology, Wroclaw, Poland. E-mail: zbigniew.palmowski@pwr.edu.pl 2 Zbigniew Palmowski (see [1,2]). Very natural generalizations can concern larger amount of parallel queues, the case where apart from the common input each queue has its own batch arrivals etc. 2 Problem statement The natural questions concern the exact form of H(x, y) and its asymptotics for large values of x and y. Exact expression for H(x, y). We have H(x, y) = ∑m>x,n>y p(m, n) for the stationary distribution p(m, n) = P(Q1∞ = m, Q2∞ = n). Moreover, p(m, n) is a stationary distribution of the discrete time Markov chain {(Q1n , Q2n ), n ∈ N ∪ {0}}, hence it satisfies a classical balance equation. This probably can be used to get H(x, y) at least in two cases. In the first case, An , Sni , i = 1, 2, are 0 − 1 random variables with P(An = 1) < min{P(Sn1 = 1), P(Sn2 = 1)}. In the second case, we can assume that An , Sni (i = 1, 2) have geometric laws, that is, P(Sni = k) = βi (1 − βi)k , i = 1, 2, k = 0, 1, 2, 3, . . . P(An = k) = α (1 − α )k , n o and 1−αα < min 1−β β1 , 1−β β2 . 1 2 Light-tailed asymptotics. Under light-tailed assumptions we look for the asymptotics of H(nη1 , nη2 ) as n → ∞, where ηi > 0, i=1,2. Heavy-tailed asymptotics. Here we assume the complementary condition that the arrival sizes An are strongly subexponential, that is, that P(A1 + A2 > n) =2 n→∞ P(A1 > n) lim n and ∑ P(A1 > n − k)P(A1 > k) ∼ 2EA1 P(A1 > n) k=0 as n → +∞, where f (n) ∼ g(n) when limn→∞ f (n)/g(n) = 1. The goal is to use the principle of a single big jump to find (under some moment assumptions put on the service capacities Sni served in one slot), the asymptotics of H(nη1 , nη2 ) as n → ∞. 3 Discussion Exact expression for H(x, y). We believe that one should start from the balance equation. For example in the second case we have m p(m, n) = n ∑ ∑ ∑ p(m − k, n − l)α (1 − α )sβ1 (1 − β1)l1 β2 (1 − β2)l2 . k=−∞ l=−∞ l1 −s=k l2 −s=l Moreover, one can use the bivariate moment generating function H(z, w) = m n ∑∞ m,n=0 z w p(m, n) to simplify above equations and try to identify the solution (see [5] and [9] for similar considerations). Light-tailed asymptotics. The case when An , Sni , i = 1, 2, are 0 − 1 random variables was analysed in [8]. For First exit time for a discrete time parallel queue 3 the case of an arbitrary direction, [2] derived similar but finer asymptotics. However, it seems that they are not fully proved. It would be nice if the conjecture can be fully proved. For this, it might be necessary to restrict the direction for the tail asymptotic because the polynomial pre-factor may be different from n−1/2 depending on the direction as studied in [7,12]. To get exponential asymptotics for general distributions of An , Sni , i = 1, 2 (for example geometric ones) one can follow [3,6,7,12]. That is, for ϕ ∈ R2 we denote ϕ (ϑ ) = E exp{< Qn , ϑ >}. We assume that there exists a solution (γ , s) ∈ R2 ×  (0, ∞) of the  ′ ′ Cramér equation: ϕ (γ ) = 1, ϕ (γ ) = η s, where η = (η1 , η2 ) and ϕ (γ ) = ∂ ∂ϕγ(γ ) , ∂ ∂ϕγ(γ ) . 1 2 Then one can conjecture that (under some additional assumptions) H(nη1 , nη2 ) ∼ Cn−1/2 e−<γ ,η >n for some constant C. One can think of identifying the constant C for some particular laws of arrival and service sizes as well. Heavy-tailed asymptotics. Assume that Sni ≥ 1, i = 1, 2. We believe that one can use an idea given in [10] representing H(x, y) as a crossing probability of an increasing and random barrier by a random walk An and then apply [11] to prove that ∞ H(nη1 , nη2 ) ∼ ∑ P(A1 > max{nη1 + kES1, nη2 + kES2}) as n → +∞. k=0 Acknowledgements The research of Zbigniew Palmowski is partially supported by Polish National Science Centre Grant No. 2018/29/B/ST1/00756 (2019-2022). References 1. Badila, E.S., Boxma, O. and Resing, J.A.C. (2015). Two parallel insurance lines with simultaneous arrivals and risks correlated with inter-arrival times. Insurance Math. Econom. 61, 48–61. 2. Borovkov, A.A. and Mogulskii, A.A. (2001). Large deviations for Markov chains in the positive quadrant. Russian Math. Surveys 56, 803–916. 3. Borovkov, A.A. and Mogulskii, A.A. (1996). The second rate function and the asymptotic problems of renewal and hitting the boundary for multidimensional random walks. Siberian Math. J. 37, 745–782. 4. Boxma, O., Koole, G. and Liu Z. (1994). Queueing-theoretic solution methods for models of parallel distributed systems. In Performance Evaluation of Parallel and Distributed Systems, pages 1–24, CWI Tract 105, Amsterdam. 5. Cohen, J.W. and Boxma, O.J. (1983). Boundary Value Problems in Queueing System Analysis. NorthHolland Mathematics Studies, Vol. 79, North-Holland Publishing Company, Amsterdam. 6. Collamore, J.F. (1996). Hitting probabilities and large deviations. Ann. Probab. 24(4), 2065–2078. 7. Dai, J.G. and Miyazawa, M. (2011) Reflecting Brownian motion in two dimensions: Exact asymptotics for the stationary distribution. Stoch. Syst. 1(1), 146–208. 8. Fayolle G., Iasnogorodski, R. and Malyshev, V. (2017) Random Walks in the Quarter Plane. Algebraic Methods, Boundary Value Problems, Applications to Queueing Systems and Analytic Combinatorics. Springer. 9. Fayolle G. and Iasnogorodski, R. (1979). Two coupled processors: the reduction to a Riemann-Hilbert problem. Z. Wahrscheinlichkeitstheorie verw. Gebiete 47, 325–351. 10. Foss, S., Korshunov, D., Palmowski, Z. and Rolski, T. (2017). Two-dimensional ruin probability for subexponential claim size. Probab. Math. Stat. 37(2), 319–335. 11. Foss, S., Palmowski, Z. and Zachary, S. (2005). The probability of exceeding a high boundary on a random time interval for a heavy-tailed random walk. Ann. Appl. Probab. 3, 1936–1957. 12. Kobayashi, M. and Miyazawa, M. (2014). Tail asymptotics of the stationary distribution of a twodimensional reflecting random walk with unbounded upward jumps. Adv. Appl. Probab. 46(2), 365–399. 13. Lieshout P. and Mandjes, M. (2007). Tandem Brownian queues. Math. Methods Oper. Res. 66, 275–298.