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This paper presents a mathematical model based on dynamic pump-wavelength selection for an optical packet switch (OPS). In the OPS, multiple packets that carry the same wavelength from different input ports could be addressed to the same output port at the same time slot. This condition is called wavelength contention. Of those contended packets, only one is forwarded to the output fiber while the others are dropped. Parametric wavelength conversion is used to convert the contended wavelengths into available non-contending wavelengths. The OPS based on the dynamic pump-wavelength selection scheme, where the pump-wavelengths are adjusted based on the requests in every time slot, uses a heuristic matching algorithm to minimize the number of packet losses. However, there is no guarantee that the heuristic algorithm outputs the optimum result. The mathematical model presented in this paper is used to confirm the performance of the heuristic matching algorithm for the DPS-based OPS. A simulation shows that the heuristic matching algorithm achieves the same performance as the optimum solution provided by the mathematical model.

Optical packet switched (OPS) networks are emerging as a serious candidate for the evolution of optical telecommunication networks needed to support high-throughput services such as voice over IP (VoIP) and high quality video streaming on demand. In an optical packet network with OPSs interconnected with optical fibers running wavelength division multiplexing (WDM), packets are transmitted from source to destination without any optical-electrical-optical (O/E/O) conversion.

In an OPS, an optical fiber carries several wavelengths to transmit packets. An input fiber may carry as many packets as wavelengths, which could be destined to one or several output fibers. In an OPS, output contention occurs when multiple packets from different input fibers share the destination of a single output fiber even though they use the same wavelength at the same time. Only one of the contending packets is forwarded to the output fiber, the others are dropped. To select packets (and wavelengths), a switch configured by a scheduler can be used. In an OPS, the set of aggregated wavelengths coming from an input fiber is optically demultiplexed into individual wavelengths and each wavelength is assigned to an input port of the switch fabric that connects outputs to inputs. The outputs of the switch fabric are assigned different wavelengths. Therefore, each input port of the switch fabric can be connected to an output port of the same wavelength. The set of individual wavelengths of an output fiber are then aggregated before they egress the switch. In the switch fabric, the scheduler decides which inputs and outputs to interconnect for each wavelength by performing a matching process (i.e., the assignment of one input to one output). The case wherein several inputs of the same wavelength have packets for the same output is said to be output contention.

Some methods proposed for resolving output contention use 1) optical buffering through fiber delay lines (FDLs) [

A practical approach to avoiding contention is to use wavelength conversion [

A parametric wavelength converter (PWC) [

wavelengths in this example,

Multiple wavelength conversion based on a parametric process appears to be becoming feasible. The simultaneous multiple wavelength conversion of over 30 channels has been reported using fiber [_{3} waveguides [

Several studies describe OPSs that use PWCs [

A heuristic matching algorithm was introduced for the DPS-based OPS [

In this paper, a mathematical model for the DPS-based OPS is proposed as a reference with which to confirm the performance of the heuristic matching algorithm in [

A dynamic pump-wavelength selection (DPS) switch, is an OPS using PWCs where the set of pump-wave- lengths is dynamically changed in every time slot based on requests. For the general case, the set of pump-wa-

velengths is defined as

transmission wavelength index for the Mth PWC.

The DPS switch, whose set of pump-wavelengths can be altered on a time slot basis, consists of three parts: the controller, the pump-wavelength generator, and the main switch (see

The controller performs both matching between input and output ports and selecting pump-wavelengths in an integrated manner on a time slot basis. It controls both the switching configuration and pump-wavelength selection by using a matching scheme. It uses electrical signals to communicate with the pump-wavelength generator and the main switch.

The pump-wavelength generator sets pump-wavelengths that are determined by the controller in every time slot.

The main switch is an OPS with N input and output fibers and M PWCs. Each fiber carries W different wavelengths. Demultiplexers at the output of PWCs are used. Each demultiplexed wavelength has a one-to-one correspondence with an input port of the switch fabric. The individual wavelengths, coming through individual input ports of the switch fabric, are grouped by an optical coupler before being forwarded to the output fiber. In

this way, each converted wavelength can be connected to any output.

A mathematical model is a convenient tool for solving optimization problems. In this paper, the problem is modeled as binary integer linear programming (BILP) to maximize number of connections between input and output ports via PWCs, which in turn minimizes the number of packet losses. In the BILP, the variables are required to be 1 or 0 rather than arbitrary integers. The terminology is as follows.

Objective function

Subject to

Equation (1a) is the objective function of this BILP model. It maximizes the number of connections between input and output ports via PWCs. Parameter

Equations (1b)-(1i) are constraints of this BILP model. While Equations (1b)-(1d) define general characteristics of PWC, Equations (1e)-(1i) define the special features of a PWC based on DPS. Equation (1b) is the constraint that limits the total number of successful requests so that it does not outnumber the total number of incoming requests. Equation (1c) is the constraint that defines that each oth output fiber at

In the simulation, the performance of the heuristic matching algorithm of OPS with DPS-based PWC is confirmed against the optimum solution provided by the proposed mathematical model, in term of packet loss rate. The packet loss rate is the difference between the total number of incoming requests and the total number of successful requests, with regard to the total number of incoming requests. An OPS with

We assume that packet arrival at

Figures 3(a)-(d) show that the performance of the mathematical model matches that of the heuristic algorithm for all parameter combinations examined.

Packet loss rates under non-uniform traffic were examined using four well-known traffic models: unbalanced [

The traffic is uniform if the destinations are uniformly distributed among all output ports [

The traffic is uniformly distributed if

The power of the two (PO2) traffic model [

The diagonal traffic [

Under hotspot traffic [

Figures 4(a)-(d) confirm that the packet loss rates of the heuristic algorithm are the same as those of the ma- thematical model in every non-uniform traffic pattern sampled.

assigned to output fibers with the same number as the input fiber. As a result, there is no contention when

This paper proposes a mathematical model to confirm the performance of the heuristic matching algorithm based on dynamic pump-wavelength selection for an OPS with PWCs. The objective function of the mathematical model is to maximize the number of connections, which in turn minimizes the number of packets lost in the optical packet switch between input and output ports. Simulation results show that the heuristic matching algorithm achieves the same performance as the optimum solution is provided by the mathematical model for every parameter combination tested under both uniform and non-uniform traffic patterns.

This work was supported in part by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (B) 23360168, and the Support Center for Advanced Telecommunications Technology Research (SCAT).