G. Fayolle and R. Iasnogorodski. Two coupled processors: the reduction to a Riemann-Hilbert problem. Zeitschrift fur Wahrscheinlichkeitstheorie und vervandte Gebiete, 47:325-351, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of the system, the conditions for steady state, the average delays etc. for a broad class of arrival processes. Generating functions are used in order to characterize the state probabilities. As often happens with queues that are coupled (Sidi and Segall [4] Fayolle and Iasnogorodski [7], Eisenberg ( 8,9] some boundary functions need to be determined during the solution process. A substantial effort is devoted in this paper for that task. In addition, we indicate some interesting properties of competing systems with a single server through an accompanying example. Several ....

G. Fayolle and R. Iasnogorodski, Two coupled processors-the reduction to a Riemann-Hilbert problem, Wahrscheinlichkeitstheorie (1979) 1-27.

....these systems will gain no multi server benefit when serving highly variable jobs; short jobs may get stuck waiting behind long jobs in the single queue for each class. All works we mention below consider Poisson arrivals. Early work on the coupled processor model was by Fayolle and Iasnogorodski [4] and Konheim, Meilijson and Melkman [6] Both papers assume exponential service times, deriving expressions for the stationary distribution of the number of jobs of each type. Fayolle and Iasnogorodski use complex algebra, eventually solving either a Dirichlet problem or a homogeneous ....

G. Fayolle and R. Iasnogorodski. Two coupled processors: the reduction to a Riemann-Hilbert problem. Zeitschrift fur Wahrscheinlichkeitstheorie und vervandte Gebiete, 47:325--351, 1979.

....of F i (x) is itself rational. This result allows us to formulate and solve a Riemann Hilbert boundary value problem on the unit circle in Section 5. The use of boundary value problems to solve queueing problems is not new and can be traced back to the seminal work by Fayolle and Iasnogorodski [12] (see also the monograph [8] and the more recent [13] as well as the references therein) Solving the boundary value problem allows us to compute the z transform of the queue length, as shown in Section 6. A word on the notation used in this paper: C will denote the set of all complex numbers, ....

G. Fayolle and R. Iasnogorodski. Two coupled processors: the reduction to a Riemann-Hilbert problem. Z. Wahrsch. Verw. Gebiete 47 (1979), 325-351.

....its algebraicity. The kernel method has been part of the folklore of combinatorialists for some time and is related to the what is known as the quadratic method in enumeration of planar maps [13] Earlier references (see [25] Ex. 2.2.1.11 for Dyck paths, 39, Sec. 15.4] for a pebbling game, [20] for a queuing theory application) deals with the case of a functional equation of the form K(z; u)F (z; u) A(z; u) B(z;u)G(z) with F; G the unknown functions) when there is only one small branch, u 1 , such that K(z; u 1 (z) 0. In that case, a single substitution does the job, and G(z) ....

Guy Fayolle and Roudolf Iasnogorodski. Two coupled processors: the reduction to a Riemann-Hilbert problem. Z. Wahrsch. Verw. Gebiete, 47(3):325-351, 1979.

....of F i (x) is itself rational. This result allows us to formulate and solve a Riemann Hilbert boundary value problem on the unit circle in Section 5. The use of boundary value problems to solve queueing problems is not new and can be traced back to the seminal work by Fayolle and Iasnogorodski [12] (see also the monograph [8] and the more recent [13] as well as the references therein) Solving the boundary value problem allows us to compute the z transform of the queue length, as shown in Section 6. A word on the notation used in this paper: C will denote the set of all complex numbers, ....

G. Fayolle and R. Iasnogorodski. Two coupled processors: the reduction to a Riemann-Hilbert problem. Z. Wahrsch. Verw. Gebiete 47 (1979), 325--351.

.... itself closely related to various queueing theory models, involves lattice paths that describe an interesting collection of events (the embedded Markov chain) The recent book by Fayolle et al. on random walks in the quarter plane [26] is historically motivated by such queueing theory questions [25]. Word representations of lattice paths also provide many examples of contextfree languages. This side of the coin is closely related to encodings of trees by words, so that Dyck paths (that are associated to general trees and binary trees) and Motzkin paths (that encode unary binary trees) play ....

....the deep rami cations of the theory in the harder case of walks in a quarter plane not satisfying directedness restrictiction (thus, a pure 2 dimensional problem) but their methods only apply to nearestneighbour moves. The book [26] itself draws some of its inspiration from the early paper [25] where a sophisticated use of the kernel method already plays a central r ole (amongst other techniques like conjugacy and Riemann Hilbert problems) see also the references to Flatto and Malyshev s works in [61, p. 1208] and the historical comments in [26, p. VII XI] 2.3. Computational aspects. ....

Guy Fayolle and Roudolf Iasnogorodski, Two coupled processors: the reduction to a RiemannHilbert problem, Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 47 (1979), no. 3, 325-351.

....the speed of server 1 is r 1 1, and if server 1 is idle then the speed of server 2 is r 2 1. In a sense, the servers are coupled, and a server with no work at its own queue is able to assist the other server. This coupled processors model has been analysed by Fayolle and Iasnogorodski [11] and by Konheim, Meilijson and Melkman [13] in the case of negative exponentially distributed service requests, and by Cohen and Boxma [9] in the case of generally distributed service requests. Konheim et al. apply the uniformisation technique; Fayolle and Iasnogorodski determine the joint queue ....

....process. The queueing analysis of GPS is extremely difficult. Interesting partial results were obtained in [2] 10] 14] 17] If N = 2, then the above coupledprocessors model with r 1 = r 2 = 2 coincides with the GPS model with equal weights; hence the exact queue length analysis in [11], 13] for the case of exponentially distributed service requests, applies to this special GPS case. Furthermore, the exact analysis of the joint workload process in [9] which holds for generally distributed service requests, is also applicable. The latter study forms the starting point of the ....

G. Fayolle and R. Iasnogorodski (1979). Two coupled processors: the reduction to a Riemann-Hilbert problem. Z. Wahrsch. Verw. Gebiete 47, 325-351.

.... itself closely related to various queueing theory models, involves lattice paths that describe an interesting collection of events (the embedded Markov chain) The recent book by Fayolle et al. on random walks in the quarter plane [26] is historically motivated by such queueing theory questions [25]. Word representations of lattice paths also provide many examples of contextfree languages. This side of the coin is closely related to encodings of trees by words, so that Dyck paths (that are associated to general trees and binary trees) and Motzkin paths (that encode unary binary trees) play ....

....systems: see, e.g. Robert s book [66] for an account. The beautiful synthesis by Fayolle, Iasnogorodski, and Malyshev [26] exposes the deep rami cations of the theory in the case of nearest neighbour walks in a quarter plane. The book [26] itself draws some of its inspiration from the early paper [25] where a sophisticated use of the kernel method already plays a central r ole (amongst other techniques like conjugacy and Riemann Hilbert problems) 2.3. Computational aspects. We discuss now a way to determine directly the equations satis ed by the algebraic functions encountered so far. Because ....

Guy Fayolle and Roudolf Iasnogorodski, Two coupled processors: the reduction to a RiemannHilbert problem, Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 47 (1979), no. 3, 325-351.

....server is idle, i = 1; 2. In a sense, the servers are coupled, and a server with no work at its own queue is able to assist the other server. Notice that r 1 = r 2 = 2 corresponds to GPS with OE 1 = OE 2 = 1 2 . This coupled processors model has been analyzed by Fayolle Iasnogorodski [25] and by Konheim, Meilijson Melkman [33] in the case of negative exponentially distributed service requests, and by Cohen Boxma [22] for generally distributed service requests. In the latter case, the joint distribution of the workloads in both queues was obtained by formulating and solving a ....

Fayolle, G., Iasnogorodski, R. (1979). Two coupled processors: the reduction to a Riemann-Hilbert problem. Z. Wahrsch. Verw. Gebiete 47, 325--351.

....In these PS disciplines all jobs receive service simultaneously, whereas in many queue head of the line PS systems only those at the head of the queues receive service. Head of the line PS for two queues and exponential service times has been analyzed by several authors. Fayolle and Iasnogorodski [FI] derived by considering two dimensional birth and death processes, albeit complicated, analytical expressions for the generating function of the queue length in the general case of two asymmetric queues covering our model, cf. the comments below and Remark 6.3. In [KMM] this generating function is ....

....distribution satisfies a functional equation, which can be transformed into a Riemann Hilbert problem. Although our particular two dimensional birth and death process and hence our Riemann Hilbert problem is a special case of the more general class of birth and death processes treated in [FI] by means of Riemann Hilbert problems, we obtain by using different constructions and exploiting the special structure of our problem more explicite results. The solution of the Riemann Hilbert problem can be reduced to a Dirichlet problem for a circle, which solution by Schwarz s formula yields ....

[Article contains additional citation context not shown here]

Fayolle, G., Iasnogorodski, R., Two Coupled Processors: The Reduction to a RiemannHilbert Problem. Z. Wahrscheinlichkeitstheorie verw. Gebiete 47 (1979) 325--351.

....the speed of server 1 is r 1 1, and if server 1 is idle then the speed of server 2 is r 2 1. In a sense, the servers are coupled, and a server with no work at its own queue is able to assist the other server. This coupled processors model has been analysed by Fayolle and Iasnogorodski [11] and by Konheim, Meilijson and Melkman [13] in the case of negative exponentially distributed service requests, and by Cohen and Boxma [9] in the case of generally distributed service requests. Konheim et al. apply the uniformisation technique; Fayolle and Iasnogorodski determine the joint queue ....

....process. The queueing analysis of GPS is extremely difficult. Interesting partial results were obtained in [2] 10] 14] 17] If N = 2, then the above coupled processors model with r 1 = r 2 = 2 coincides with the GPS model with equal weights; hence the exact queue length analysis in [11], 13] for the case of exponentially distributed service requests, applies to this special GPS case. Furthermore, the exact analysis of the joint workload process in [9] which holds for generally distributed service requests, is also applicable. The latter study forms the starting point of the ....

G. Fayolle and R. Iasnogorodski (1979). Two coupled processors: the reduction to a Riemann-Hilbert problem. Z. Wahrsch. Verw. Gebiete 47, 325-351.

....to standard problems of the theory of boundary value equations (Wiener Hopf, Dirichlet, Riemann, Riemann Hilbert) and singular integral equations. Pioneering papers are those of Eisenberg [55] transforming a two queue polling problem into a Fredholm integral equation) and Fayolle Iasnogorodski [57] (transforming a problem concerning two processors with coupled speeds into a Riemann Hilbert boundary value problem) A systematic and detailed study of the boundary value method is presented by Cohen Boxma [43] with applications to various queueing problems: a two queue polling problem, the ....

G. Fayolle and R. Iasnogorodski. Two coupled processors: The reduction to a RiemannHilbert problem. Z. Wahrsch. Verw. Gebiete, 47:325--351, 1979.

....; N , for which the average delay in all of the queues is finite. With Bernoulli arrivals (or any other independent, identically distributed arrivals) this problem can be accurately modeled in a straight forward way as an N dimensional random walk. From the early work by Fayolle and Iasnogorodski [77] to more recent works by Szpankowski [78] Sidi and Segall [79] Rao and Ephremides [80] Anantharam [81] and others, it has become well known that such chains cannot be easily solved. Thus, much of the work has concentrated on obtaining outer and inner bounds to the region of stability. A key ....

G. Fayolle and R. Iasnogorodski, "Two coupled processors: the reduction to a Riemann-Hilbert problem," Wahrscheinlichkeitstheorie, pp. 1-27, 1979.

No context found.

G. Fayolle and R. Iasnogorodski. Two coupled processors: the reduction to a Riemann-Hilbert problem. Zeitschrift fur Wahrscheinlichkeitstheorie und vervandte Gebiete, 47:325-351, 1979.

No context found.

G. Fayolle and R. Iasnogorodski, "Two coupled processors: The reduction to a Riemann--Hilbert problem," Wahrscheinlichkeitstheorie, pp. 1--27, 1979.

No context found.

G. Fayolle and R. Iasnogorodski, "Two coupled processors: the reduction to a RiemannHilbert problem," Z. Wahrscheinlichkeitstheorie, vol. 47, pp. 325--351, 1979.

No context found.

G. Fayolle and R. Iasnogorodski, "Two coupled processors: the reduction to a Riemann-Hilbert problem," Z. Wahrscheinlichkeitstheorie, vol. 47, pp. 325--351, 1979.

No context found.

G. Fayolle and R. Iasnogorodski, "Two coupled processors: the reduction to a Riemann-Hilbert problem," Z. Wahrscheinlichkeitstheorie, vol. 47, pp. 325--351, 1979.

No context found.

Guy Fayolle and Roudolf Iasnogorodski, Two coupled processors: the reduction to a RiemannHilbert problem, Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 47 (1979), no. 3, 325-351.

No context found.

Guy Fayolle and Roudolf Iasnogorodski. Two coupled processors: the reduction to a Riemann-Hilbert problem. Zeitschrift fr Wahrscheinlichkeitstheorie und Verwandte Gebiete, 47(3):325351, 1979.

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