In the past twenty years, many web based companies have been created to allow, facilitate and service the large and growing number of customers’ interactions. Successful organisations such as Google and Facebook have several billion active users [1] [2] . Service provision in advanced economies has grown substantially in significance. The resource needs and implications for the developing service need to be predicted. To make an organisation functionally efficient and profitable, it is critical to be able to predict staff allocations and productivity required for the business model [3] [4] [5] [6] [7] . System Dynamics (SD) models were first used to facilitate these computations [8] . Hajiheydari and Zarei [9] use SD models to inform the Business Model (BM) decision making process, later evaluated by expert opinion. Little or no examination has been made of such businesses in a dynamic sense.

A service economy such as those in the web based environment is seen as different from a product dominated economy. Service Science is a branch of Systems Theory [10] which combines engineering with management and social science, provides a theoretical basis for this change in emphasis. In a rapidly changing world, models of the development of services are required, to enable companies and managers to compare their performance with what is possible and practicable. An example is the development of software [11] . These models have been used to estimate the effects of rework as an important component of the total effort needed. The value that a customer sees in the service provided by a business is a key issue in how the company subsequently develops throughout time.

The aim of this paper is to build and evaluate a control engineering based model of value development with external and internal actors. The results of a case study using computer simulation of the model with SIMULINK are then compared to the published development of the Wikipedia web based system to determine the validity of the model.

Current theories of value generation specify the inclusion of the concept of co-creation of value by customers. Use of a large number of co-creators makes managing the complexity of the value generation process much more difficult. The problems of using numbers of co-creators or editors in the Wikipedia environment have sparked considerable research into the reasons why people contribute in this way [12] . Their principle findings are related to motivation and to the problems of conflict. Similar statistical modelling work on motivation of open-source software contributors by Santos et al. [13] has shown that the attractiveness of the work is the most important factor and this can limit participation to only a few contributions.

Few reliable SD models have been derived for the case of web-based co-development and as yet no fully validated models of any type are available. It is this issue that this model and validation application addresses. Current literature is now reviewed to show where the present application fits. This is followed by a description of the methodology used and the results obtained.

Forrester [14] discusses the lessons from System Dynamics (SD) modelling for business and social applications stating that the purpose of such models is to “organize, clarify and unify knowledge” “of an important system”. To model the collaborative development of a web service, the various actors and users of the system, as well as the underlying dynamics of the system, have to be taken into account. Existing models of web services are explored to see the range of applicability and then the current viewpoints in value estimation techniques are critiqued.

System Dynamics was developed from control theory by Forrester [8] in the 1950’s to enable those methods to be applied to non-engineering situations. These SD models use the basic principles that all systems have feedback, both positive and negative and suffer delays in information processing or physical flows.

Control System (CS) models have been applied to software development by the first author [52] to determine project stability and optimise agile methods, [53] , to supply chain behaviour by the Cardiff School [54] and labour supply by the current authors to see what effects the addition of labour has to simplest supply chain [55] for the stability of such systems. Such models are useful because no human devised system is in equilibrium, as is assumed by SERVQUAL and SERVPERF, but is a dynamic time dependent process. It is certainly clear from the published Wikipedia results [56] that this is the case. Their advantage over SD models is the facility to vary the type of delay, nonlinear functions and variable gain and produce optimal solutions. These models allow researchers to investigate the causes of such dynamic behaviour in a wide range of problem areas. The fundamental purpose of such models is to identify feedback processes, effective time delays and amplifications of signals that dominate the dynamics.

As indicated by the survey of feedback by Richardson [28] , processes in this field contain feedback loops and recognition of a deficit between a desired state and the actual state. It is worthwhile noting that real physical control processes in biology use difference measurements. All the models of software processes examined by Abdel-Hamid [16] are represented using this structure with considerable success. The SERVQUAL representation (fig 1) echoes this but it is not the rationale for the choice of this model structure, rather the model provides evidence for some of the basic structure of SERVQUAL assessment but not its measurement processes.

The prime purpose of a service value model is to enable the development process to be predicted and hence, controlled or moderated in an inclusive and collaborative development environment where no one person or organization has sufficient total control authority. This model relates to the parameters influencing co-creation of service with assumptions that include co-creation (or co-production) as a normal activity by originators of the articles and editors. The exceptions are that the productivity of the internal and external sources may not be the same, and the associated time delays may be different, usually slower for the external contributors. It is assumed here that the users have already chosen to use “Self-Service Technologies”, which is an issue discussed in some detail by Meuter et al. [57] . They outline the problem of adoption of such new technology discussing the ability, motivation and clarity of role of users. These are well defined here because of the specific nature of the Wikipedia example but the agent classification would need to be made clear if the model is used elsewhere. The model does not have these parameters deployed separately but they could be a prime example of such an improvement or extension. Both users (co-creators) and editors are largely self-selecting in the Wikipedia example. This will not always be the case.

To formulate the model of service development, terms need to be carefully defined. Service value is defined here as a state. This is a key concept. If value can be measured in any concrete form then it can be defined as a state since it can be added to and can be depleted. In the case of Wikipedia the value (state) is measured as numbers of completed tasks or articles added to the encyclopaedia. The more articles there are the more use (value) the encyclopaedia has to the end user. In the case of open-source software development the value (state) would be some function of the number of lines of code implemented or task functions completed.

The overall structure of the model with subsystems is shown in a causal loop diagram in Figure 2 and the Simulink diagram Figure 3. In the model, Value (number of articles) is increased at a rate of AVR and destroyed at a rate of DVR [58] . In all projects there is some rework. Open-source software development follows the same patterns as for in-house developed software, except that timescales are likely to be much more protracted since there is no control of when external agents do their work. In the case of Wikipedia, articles are removed if they are judged to be poorly drafted, incorrect or malicious [59] . This type of process is seen in many natural and manmade systems [58] and is a common feature of SD models.

The rate of change of the state, (Actual Value AV), equals the difference between the rates of change of value created (AVR) and the rate of change of value

destroyed (DVR) (related to rework). This is a very common formulation in SD and the equation relating such processes can be written:

d A V d t = A V R − D V R (1)

In the centre of the figure the difference between AVR and DVR is integrated to obtain the actual value, AV. The rate of value destroyed is dependent on the number of agents (editors) at any time t. There will be a sudden change in the value destroyed as the editors become aware of the submission of material (articles). This is obtained from the number of agents (editors) multiplied by the input step demand then delayed by a time T_ad, the time taken for the material (documents) to be removed from the system represented by the control system element: 1/(Tdvs + 1) where s is the Laplace transform. The amount of rejected material depends on the fraction of the original amount (number) of material (documents) and the average productivity of the agents on average. The output of the delayed block is then multiplied by the fraction of the original documents rejected and multiplied by the average productivity of the agents, pre.

These time delays are observed and recorded in practice. However the form of modelling here is the simplest formulation that could be used. The current data available does not allow for a higher order model interpretation.

The value generated rate AVR, is made up of two contributions, from internal and external contributors. Both arise from the perception of a deficit or gap in service after a time T_p. The internal component is due to the number of internal staff N_e, multiplied by their productivity pre (Gain 7).

The external component is due to the number of external editors multiplied by their productivity. An assumption is made that their productivity is essentially the same, pre, as the internal staff (Figure 4).

In the development of a web-based service, such as open-source software production, it is often the case that internal staff are used to check the material added to the overall value rather than create new value. These contributions are added and delayed by time T_av, which is the time taken to accept new material. This is also publically available data. Gaps 5 & 6 are implicitly embedded in this module.

The form of service value is defined in the previous section by a perception of a service gap pgV, is equal to the expected value minus the actual value, however defined or measured (e.g. price). This is illustrated in Figure 5. There are again two parts to the perception of value, that arising from an expectation by users that changes with time and can be implemented as a linear function and a second part where this expectation is generated by the editors. These perceptions in principle match SERVQUAL Gaps 1, 2, 3 & 7 in Figure 1.

p g V = e V − A V (2)

The number of contributors or participants in the development of value will grow, exponentially, as the word of the opportunity to contribute spreads. Ultimately this will reach a limiting number of capable contributors, co-creators,

either because they do not have the time, expertise or because they get disillusioned with the exercise [60] . Suh et al. [61] point out two other issues that could cause this problem of a downturn in output. These are growing resistance to new content by the small number of prolific editors and the greater overhead imposed by the costs of coordination and bureaucracy.

They also raise the issue of vandalism on various topics, which is topic specific. The rate at which informed participants increases depends on the number of participants who spread the word [62] .

d N d t = A N (3)

As a first approximation it is assumed that the reduction will also be a linear rate. This value has been approximated from known data.

after a Time delay T_nf to disengagement this then reduces at rate edr

This is implemented in Figure 6.

All data for use to validate the model was derived from the data published online in Wikipedia [56] [58] [63] [64] . The form of this model is similar to that for simplified software development processes [52] . The SERVQUAL description of Figure 1 does not match exactly the CS model but gaps or differences in perception can be identified as causal elements in the creation of value process. In order to evaluate the model to see if its’ features are valid it has been used to determine the development of Wikipedia articles with time as indicated in Section 4.

The published Wikipedia data used for the case study simulation is given in Table 2. Generally, the published response [63] (Figure 7) for the production of Wikipedia articles shows a rising trend, at first nearly exponential, but tending towards a linear increase with time after 5 years, with a number of oscillations present in the curve of period around 2 - 4 months.

There is a small surge in articles in the real data at around 18 months from Wikipedia’s start but the cause is unknown, contributors hint that it may be due to a statistical fault. The increase in the number of editors (contributors) of articles increases exponentially but reaches a limit and then declines. There is a substantial reduction in trend at around 80 months. It is not known exactly what has caused this change.

The latter part of the data exhibits larger (up to 2000 editors) oscillations of around 3 - 4 months period. Results from the simulation in Figure 8 show similar shaped curves to Figure 7 with relatively good numerical agreement.

The curves show an exponential rise to 5 years for both articles produced and the number of editors and the magnitudes are close using the best estimates from published wiki data [63] .

The discontinuity in the simulated data is sharper than in the real data. The editor simulation does not show the oscillations that are present in the real data.

There are oscillations in the curve, for the articles produced after 5 years, but they are smaller than for the real data. No substantial oscillations are present in the simulated editor growth curve.

“WMFArticlesVsContrib” by Bridgestone Partners [63] a consultant to the Wikimedia Foundation, Licensed under GFDL via Wikimedia Commons https://commons.wikimedia.org/wiki/File:WMFArticlesVsContrib.png#/media/File:WMFArticlesVsContrib.png

Figure 9 shows the effect of changes in contributor productivity the dominant parameter in the performance of the model and a similar but smaller effect of the productivity of the editors in Figure 10.

A plot of the productivity of editors shows a similar but smaller effect. Productivity of the contributor’s increases the number of articles produced as would be expected but the initial number of editors also increases the production of finished articles (Figure 10). An increase in the fraction of articles rejected by

editors to 30% has a significant effect as shown in Figure 11, reducing the number of articles completed by around 5%. A reduction of 12% in initial editor numbers (Figure 12) produces a similar reduction in articles generated, while a 30% reduction in initial expected articles produces a 5% reduction in articles produced (Figure 13).

These observed patterns are visible in the articles in other languages as well but are much more defined in English since there are a greater number of articles involved. The oscillations could be associated with US academic semesters with peaks in February and August. Research reported by Faulkner et al. [65] appears to indicate that the decline in number of editors is due to a rate of decrease of new editors.

A recognition of a defined gap in service would tend to support the SERVQUAL description, but not the methodology to obtain an estimate of value. This agreement supports both a recognition of a gap in service and the basic dynamics as implemented in equation 1. That there is a limit followed by a slow decline as applied to the case of Wikipedia is confirmed. However, it is also clear that not all the dynamics of the process are included in the model and the time constants need better refinement and formulation to get the oscillations in the article generation correct. It must be clearly understood that the data used in the model are average values as presented in the literature. A separate sub-system could be included, as in software generation SD models, to allow for editor training and levels of experience to get more accurate modelling of the processes involved.

It is clear that the model does produce a good simulation of the general behaviour of the value generation process, within 10% for the growth in articles and within 25% for the editor data curves, especially reproducing the rapid exponential rise in articles.

Such a close matching result, in our experience, is very good for non-engineering applications.

The oscillations in the editor contribution are more difficult to explain. No data yet found gives any reason that could cause this small but definitive cyclic effect. It may be involved with information delays or the time delay to get approval for the removal processes but on available data these do not look likely. It is more likely that some underlying feedback paths have not been identified.

The sharp discontinuity or limit in the control model is due to the very simple implementation used here. Modelling the relevant feedback process of which the editors are part, has not yet been achieved by using a variety of simple models with suitable time constants. The way in which the expected initial value is implemented in the model is also in need of improvement, but no survey data available is sufficient to refine this part of the model.

The model, which is similar to SD project models is applicable in diverse situations such as social housing and production inventory and now proposed for value generation. In general the overall generation of value and destruction of value Equation (2) is supported by the Wikipedia simulation as the values are within the margin for the range of the constants used. That there will be a limit in production of value is supported but the implementation mechanism needs to be improved.

A control system model has been used to predict service value development using the concepts of value gaps, feedback and co-creation or co-development processes in the case of Wikipedia. The control engineering model presented here, although relatively basic, produces curves that are very close to the actual increase in articles and editors for the Wikipedia website using data publicly available from Wikipedia.

Although the model supports some of the gaps proposed in SERVQUAL, the dynamics are not captured by that description. The way the data is obtained is not useful for the creation and support of numerical models, even if that methodology is correct, which is disputed by many researchers.

The model is comparable to published data and reasonably accurate in time and scale while not representing all the minor dynamics present in the real-world situation. It would take a substantial research programme to obtain sufficient data to investigate the minor dynamics present in the real data.

Nevertheless, the basic model presented here is accurate enough to enable a company to use it to predict personnel planning and productivity necessary for task completion for other similar applications.

Gap concepts of SERVQUAL are largely but not wholly coincident with the model here. Differences in states are fundamental to the operation of SD/Control System models. The results of the model should be seen as supporting SERVQUAL not the other way around. Generally, SERVQUAL and its rivals are of little help with creating predictive numerical models.

Future development of the model depends on more information becoming available.

The application study presented in this paper should be considered as the minimum CS model that represents the fundamental development of value generation in collaborative web projects such as Wikipedia.

This CS model could be applied to open source software development and with other collaborative web development since the principle components of value generated and rework are similar in all software development.

Future research would include modelling different implementations of saturation limits and the small cyclic effects discussed earlier. Attempts are underway to model the generation of open software but have been limited by lack of data thus far.

The current simulation CS model supports the incorporation of time dependent parameters and service gap factors to assist in predicting or deriving service value development.

The authors declare no conflicts of interest regarding the publication of this paper.

White, A.S., Nielsen, D. and Censlive, M. (2019) A Control Theoretical Model of Web Service Value Development. Open Journal of Modelling and Simulation, 7, 125-147. https://doi.org/10.4236/ojmsi.2019.72007

A = fraction of population inclined to contribute

AVR = F_a (time, number of contributors, quality of contributions)

DVR = F_d (time, ideas too complex to deliver, number of contributors, quality)

Edr = disengagement rate

eV = expected value

eVd = eV − AV

eV_i = initial expected value

eVs = rate of increase of expectation

pgV = perceived value gap

E = expectation of each contributor

fr = fraction of articles rejected by editors

N = number of contributors

N_e = number of employees

pr = productivity, suffix e editors c contributors

T_av = time delay in implementing the rate of adding value

T_p = time delay in perceiving the value gap

T_dv = time delay in implementing the destruction of value

T_ev = time delay in estimating the expected value

T_nf = time to fatigue