Unlike the player patience data, session times can not be fitted with a simple negative exponential distribution. However, the data can be closely matched to a Weibull distribution, a more general distribution that is often used to model lifetime distributions in reliability engineering . Since quitting the game can be viewed as an attention ``failure'' on the part of the player, the Weibull distribution is well-suited for this application. The generalized Weibull distribution has three parameters , , and and is shown below.
In this form, is a shape parameter or slope of the distribution, is a scale parameter, and is a location parameter. As the location of the distribution is at the origin, is set to zero, giving us the two-parameter form for the Weibull PDF.
Using a probability plotting method , we estimated the shape () and scale () parameters of the session time PDF. As Figure 2(a) shows, a Weibull distribution with , , and closely fits the PDF of measured session times for the trace.
This result is in contrast to previous studies that have fitted a negative exponential distribution to session-times of multiplayer games . Unlike the Weibull distribution which has independent scale and shape parameters, the shape of the negative exponential distribution is completely determined by , the failure rate. Due to the memory-less property of the negative exponential distribution, this rate is assumed to be constant. Figure 3 shows the failure rate for individual session durations over the trace. As the figure shows, the failure rate is for flows of shorter duration, thus making it difficult to accurately fit it to a negative exponential distribution. While it is difficult to pinpoint the exact reason for this, it could be attributed to the fact that Counter-Strike servers are notoriously heterogeneous. Counter-Strike happens to be one of the most heavily modified on-line games with support for a myriad of add-on features [24,25]. Short flows could correspond to players browsing the server's features, a characteristic not predominantly found in other games. As with player patience, it may be possible to fit a negative exponential for longer session times. As part of future work, we hope examine this as well as characterize session duration distributions across a larger cross-section of games to see how distributions vary between games and game genres.