The Use of Simulations to Model Queueing Network Systems

Physical real-world phenomena can often be described by a set of equations that describe the underlying processes. A mathematical model is essentially an approximation of a real process [Bhat, 2008]. Since the analytical solution of a mathematical model is not always feasible, numerical techniques are often employed to provide realistic, but approximate solutions. Models implementing such techniques are tractable usually only if simplifying assumptions are made [Bose, 2002]. Besides the fact that numerical modelling might become intractable for complex systems, solutions provided are likely to be less accurate, due to the fact that complex systems lead to complex numerical models, which are more susceptible to computational errors [Bhat, 2008]. Instead, computer programs may be used that can mimic (i.e. simulate) the behaviour of a system more realistically (compared to numerical models), thus providing more accurate solutions regarding the behaviour of the simulated system.

The basic ideas behind developing a simulation model for a real system are illustrated in Figure 33 [Bose 2002], which shows the relation between a real physical system and the equivalent simulation model.

Figure 33: Simulating a real physical system [Bose, 2002]

In order to make a simulation model as realistic as possible, as many features and details of the real system as feasible ought to be captured. However, this is not always possible, or even practical to capture all the details of a real system in constructing the simulation model. When building a simulation model, it is advisable to make compromises regarding the incorporation of only those aspects of the real system (i.e.

model parameters) that are thought to be pertinent to the goals of the study. Hence, the

126 model parameters need to be carefully selected in order to satisfy the simulation requirements (i.e. meaningful simulation) [Bose, 2002].

The accuracy (disregarding computational errors) of a simulation model depends on how clearly the modelled system is understood, the interactions within it and hence the quality of the developed software [Bhat, 2008]. A real system has its own entities with their respective attributes that might interact with each other (i.e.

interdependencies). An entity is defined as an object of interest in a real system, such as in the simulation of a queueing network, where the individual queues and the paths followed are important functional entities. The attribute of an entity is defined as the relevant property that it is desired to study through the simulation, such as the length of queues, or the time required to serve a customer (i.e. including time spent at queue) [Bose, 2002]. In order to track a system’s state, before the initiation of a simulation model, variables are set to define the system’s parameters, including the service rates.

Finally, changes in the system can be tracked by the employed state variables that model the parameters the user is interested to study, such as the number of customers waiting at each server, or the total time taken to serve a customer [Bhat, 2008] [Bose, 2002].

The time increments in a continuous time simulator could be set arbitrarily small, if the simulator allows, thus entailing a simulation that effectively looks like a continuous time simulation. Generally, whether the state of a system can be considered continuous or discrete, and hence whether the system is a continuous or discrete time, respectively, strongly depends on the nature of the physical system modelled.

Queueing systems are examples of discrete time systems, where changes (i.e. events) in such systems happen only at specific (i.e. discrete) time instants, which are normally referred as simulation or event times. Hence, queueing system simulators are discrete time simulators that trigger their internal simulating actions only when events affecting the system’s state take place [Bose, 2002]. An event is seen as a point (i.e.

discrete) in the time (i.e. continuous) when the state of the modelled system changes.

For a queueing system an event can be the start or the end of a service [Bhat, 2008].

Discrete time simulators keep a "master event list" of the events (sequentially) that are scheduled to happen at the instants when they are going to happen. All arrival events to the service facilities of the simulated system are set at the beginning of the

127 simulation [Bhat, 2008]. Once an event at the top of the event list is being successfully processed, the simulation time increments to the time of the next event in the list happening that is then processed [Bose, 2002]. Generally, queueing simulations based on next-event incrementing procedure require fewer iterations to cover the same amount of simulated time than the fixed-time one [Hillier and Lieberman, 2001].

The best way to assess how a system behaves would be to construct a prototype model and study its behaviour (i.e. exact performance) [Hillier and Lieberman, 2001].

However, this is not usually feasible, due to time and cost limitations, especially for large complex systems. Consequently, the choice is whether numerical or simulation models best describe the physical system. The following points listed ought to be considered in making a balanced decision [Bose, 2002]:-

 Simulations are generally more realistic (i.e. closer to the real system) than numerical models, since they typically require fewer and less extreme simplifying assumptions than the latter. Also, simulations could provide crosschecks on the results provided by numerical models (i.e. verification through simulations), and can be used to assess the validity of any implied assumptions adopted in any numerical analyses;

 Simulations may allow the study of the behaviour of any attribute of any entity in the system, by simply monitoring those during the simulation runs, which would not be available from numerical models;

 A good simulation model is able to mimic a real system realistically, but would typically take long time to construct and run. The fact that simulators are computationally expensive constitutes the significant drawback that inhibits the use of simulators over simpler numerical models;

 A simulator would typically provide as outputs the moments of the parameters under interested and hence monitored during the simulation (1^st moment refers to mean value, whereas 2^nd moment refers to variance). However, unlike numerical modelling, simulations can also generate time series of selected parameters of interest, in order to demonstrate the way these change during the simulation process, in case this is of particular interest for the design of a queueing system;

128

 One of the major strengths of a numerical model is that it can abstract the essence of a problem and reveals its underlying structure, thus providing insights regarding the cause and effect relationships inside the system.

Consequently, if one is able to construct a numerical model that is both an acceptable and reasonable idealisation of the physical real problem, as well as amenable to a solution, this approach is usually preferred to a simulation.

However, many problems are too complex to be modelled numerically and simulation often constitutes the only practical approach [Hillier and Lieberman, 2001];

 Animation capabilities for displaying simulations in action can be also developed by simulators, thus enabling the user to better understand the behaviour of a modelled system. Visualisation of the operation of a system could validate a simulation model [Hillier and Lieberman, 2001];

 Neither a numerical model, nor a simulation of a physical system are likely to produce exact values for the measures of performance of the particular system, since such approaches are approximations of the real system. However, simulations are generally much better approximations of a real system, hence seen to produce better solutions [Hillier and Lieberman, 2001].

By investigating the performance of a system for a number of alternative options, one can evaluate and compare these options before narrowing down [Hillier and Lieberman, 2001]. A queueing network system can be either modelled or studied through algorithms (i.e. mathematical-numerical modelling) or through simulations.

The best course of action is usually to employ a combination of both of these approaches if applicable [Bose, 2002]. Given the limited access to commercial simulation software, the option of analytical modelling a queueing network system, as described in Sections 3.3 and 3.4, was employed to analyse the proposed network of UXVs fleet-mothership. This is a sensible, given the "Pre-Concept" nature of the research, without any current formal requirement for a UXV mothership. However, for a more formal concept phase for a UXV mothership a combination of both numerical modelling and simulation techniques would be preferable, as simulations could be conducted to refine subsystems options. In the concept phase mothership design options and alternative solutions would be considered at a sufficient, but not a totally comprehensive level of detail. One (or occasionally two) options should be

129 selected at the end of the concept phase to be subsequently worked up in later design stages, with more resources employed. The selected option would then be developed to a deeper level of detail in assessing its technical feasibility and beyond in working up the production definition. Simulation environment has been employed, for instance, in (i) modelling ship air interface framework, predicting ship-helicopter operating limits for recovering the aircraft onto the ship for new ship designs, (ii) replenishment at sea, [predicting the behaviour of ships alongside when mechanically coupled by a solid transfer system and (iii) NATO submarine rescue system, predicting the recovery system’s behaviour in high sea states [McTaggart et al, 2018] [Henry et al., 2009].

130