Applying Queueing Theory in Network Modelling

A queueing system is normally described by [Bose, 2002]:-

 Interarrival time, i.e. it describes the time between arrivals at the queue;

 Service time, i.e. this describes the size (i.e. duration) of jobs undertaken at a service node;

 Number of servers at the particular service node;

 System’s capacity, i.e. gives the maximum number of customers in the system, including both the ones currently being served and those waiting for service;

 Service discipline, i.e. this describes the service rule according to which the customers are selected to be served, such as FCFS (First Come First Served), or IS (Infinite Number of Servers).

There are two numerical methods described in the bibliography for modelling and analysing queueing networks, namely the Convolution algorithm and Mean Value Analysis (MVA) algorithm [Bose, 2002]. The Convolution algorithm is the more complex algorithm and consequently computationally more difficult, compared to the MVA algorithm. Its computational complexity increases rapidly with larger networks

107 and larger population of circulating customers (i.e. UXVs), thus such an algorithm is likely to be more susceptible to numerical accumulated errors for larger systems (i.e.

numerical errors that are carried on through the sequence of calculations as indicated by the algorithm and hence accumulated by the end of the process). However, although the Convolution algorithm is more susceptible to numerical errors, the results obtained from these two algorithms do not significantly differ. This can be seen by analysing a network, such as that shown in Figure 27, where four customers circulate through the depicted network system consisting of three single server nodes. The results, also presented in Figure 27, show the throughput "λ", which is an indication as to how fast (or slow) is the system’s performance under study (i.e. measure of number of queueing and service activities per unit time) [Bose, 2002]. It can be seen from the results presented that these two algorithms give are comparable for such a simple network system.

Figure 27: Comparison of Convolution algorithm and MVA [Adapted from Bose, 2002]

Modelling the behaviour of complex network systems using numerical techniques allows the comparison of alternative networks (i.e. options/solutions) efficiently, subject to the validity of the input parameters provided by the user (i.e. ship designer in this instance). Although the activities of a USVs fleet during a mission scenario can be quite complex, a queueing network system, such as that suggested in Figure 23, could capture the fundamental process of the overall UXVs fleet-mothership operations, by modelling the behaviour of the system and thus giving valuable information about the nodes’ functionality from the model outputs. Therefore, such a model can indicate potential areas of difficulties, i.e. underperformance of nodes according to the designer’s judgement based on potential mission related requirements (i.e. direct or proxy operational effectiveness indicators), and subsequently point out potential solutions and improvements. Consequently, given the QT capabilities on

108 network applications and the relevant information that can be extracted from such models for the design of a potential mothership, it was decided to model the behaviour of the proposed UXVs network of activities using the MVA algorithm. This decision was based on the relative simplicity of MVA algorithm, compared to the Convolution algorithm, and also on its proven robustness regarding the calculating measures of performance, as seen in the example of Figure 27. The MVA algorithm for modelling a queueing network system was implemented in FORTRAN, given the strong performance of the programming language in numerical problems, and also the author’s relative familiarity with this language. A more comprehensive description of the MVA algorithm is provided in Appendix 6, which describes:-

i. Equations and sequence of calculations;

ii. Variables (i.e. information the algorithm is imported with);

iii. Metrics that can be extracted from it (i.e. quantified measures of performance for each node in the network system).

MVA is a numerical algorithm that uses the recursive technique to obtain the transient, i.e. m = 1, 2, …, M-1 and final state, i.e. m = M, performance measures for each node of a network system, where M represents the number of UXVs in the fleet. For a given number of UXVs (i.e. M), these measures describe:-

i. Node throughput, which is rate of UXVs queueing and service activities processed at a particular node per unit time;

ii. Node processing time, which is the total time taken (i.e. actual service time plus queueing time) to serve the customers seeking a particular service;

iii. Queue formation/length, i.e. number of customers in the queue.

For a network system consisting of k, k = 1, 2, …, K, nodes (i.e. UXV tasks), with m, m = 1, 2, …, M, customers (i.e. UXVs) present, the MVA algorithm works recursively.

This means that the algorithm initiates performing the relevant calculations, by starting with zero customers in the network and incrementally calculating the performance measures of the nodes in the network system. This is done so that, as the customer population increases, by increments of one at a time, the predefined desired maximum number of customers (i.e. M) in the network is reached [Bose, 2002] [Cooper, 1981].

109 Figure 28, shows schematically the structure of the QT tool developed in FORTRAN, where the input parameters and the metrics the tool provides, are also described. Figure 29 gives the logic of the queueing network tool developed (the full coding produced is given in Appendix 6). The code consists of three distinct parts:-

i. Input file, in the form of "txt" file, that describes the necessary information fed to the algorithm, i.e model inputs in Figure 28;

ii. Actual Fortran code implementing the queueing network algorithm;

iii. Output file, in the form of "txt" file, which provides the information resulting from analysing a queueing network, i.e. model outputs in Figure 28.

110 Figure 28: Structure of FORTRAN code modelling queueing networks that represent UXVs operations supported by a mothership,

using the MVA algorithm

111 Figure 29: Flowchart of the queueing network tool developed (see Appendix 6 for

actual coding)

112

3.4 Application of Queueing Network Tool in a Mothership Concept