Model specification

To specify a queueing model, the six characteristics described in Section 3.3.2 need to be known or specified. It is reasonable to make assumptions that are conceptually compatible with the situation of visual search whenever possible and to choose the simplest option if little information is available about the appropriateness of any option.

3.3. MODEL ASSUMPTIONS AND MODEL SPECIFICATION 57

3.3.4.1 Interarrival times and service times

Although an asynchrony in the arrivals is plausible since stochastic processing time is common in signal processing systems, little is known about the time intervals between the arrivals of stimuli so far. Similarly, there is no clear empirical evidence supporting a specific probability distribution of the time required for feature integration.

Existing models for visual search (especially computational models) that involve the identification of individual stimuli are mostly based on a concrete model of the identification stage. For example, GS (Wolfe, 1994a, 2007; Wolfe et al., 1989) bases the discrimination on Signal Detection Theory (SDT; Green & Swets, 1966; cf. Cameron et al., 2004; J. Palmer et al., 2000; Verghese, 2001); the CGS model (Moran et al., 2013) on an accumulation model; Donkin and Shiffrin (2011) on the Linear Ballistic Accumulator model by Brown and Heathcote (2008). Under such models, the distribution of the identification times (corresponding to the service time in my model) may be derived. It is of course desirable to have a concrete model for the identification stage, yet the distribution derived from such a commonly used model (e.g., Wald distribution) is usually too complex as the service time distribution for a queueing model. As mentioned in the previous section, my model does not specify any concrete mechanism within either of the two processing stages. Hence the assumptions on interarrival time and service time are not restricted to a probability distribution derived from a specific mechanism.

Therefore, I opt for simpler assumptions. Under the assumptions of indepen- dent and identically distributed exponential interarrival times and service times, the queueing model has a particularly simple form: The number of customers in the queue at time 𝑡 as a random process is a continuous-time Markov chain. Restrictive as it might seem, the assumption of a exponential distribution turns out to be able to explain various temporal phenomena in nature adequately, and it is especially useful in the contexts where queueing models are applied. Another advantage of exponential distributions is that only one parameter is sufficient to characterize the probability distribution.

Specifically, I assume that

𝐴_{𝑖 ∼ Exp(𝜆), and 𝑆𝑖 ∼ Exp(𝜇)}

where 𝐴𝑖are the interarrival times and 𝑆𝑖 the service times, as defined in Section

3.3.2, and𝜆 and 𝜇 the rate parameter, respectively. In case of infinite customer source, 𝜆 and 𝜇 equal the interarival rate and the service rate, respectively. Queueing models with independent, exponentially distributed interarrival times and service times are called Markovian queueing models.

3.3.4.2 Customer population

One special characteristic of the application of a queueing model in the context of visual search is the finite customer population with non-recurrent demand, as discussed in Section 3.2. The reason is that the number of stimuli to be searched is finite and each stimulus is assumed to be searched only once at most. The assumption of finite customer source has an important impact on the arrival pattern. If the queue has an infinite customer source, the assumption of exponentially distributed interarrival times with a rate parameter𝜆 is equivalent to the statement that the arrival process is a Poisson counting process with rate 𝜆. Without an infinite customer source, the arrival process cannot be a Poisson process because the number of incoming customers has an upper limit.

In queueing theory, the arrival pattern of a finite source queue with recurrent demand has the property that the effective arrival rate is proportional to the number of customers outside the system, i.e., those who can potentially arrive in the future (see Appendix A for more details). For my model, i.e., a finite source queue with non-recurrent demand, a similar characterization of the arrival pattern is assumed. Specifically, the effective arrival rate2 is assumed to be proportional to the number of customers that have not been in the system (i.e., in the queue or at a server) yet. This means that the mean interarrival time of the next arrival is inversely proportional to the number of customers that have not yet arrived.

2_{Strictly speaking, the term “rate” should not be used here because the process is not}

stationary. The “effective arrival rate” here is to understand as the probability that an arrival occurs in the next infinitesimal time interval, see Appendix A.

3.3. MODEL ASSUMPTIONS AND MODEL SPECIFICATION 59

This characterization is assumed because such a arrival pattern is consistent with the theoretical concept of a preattentive stage. A parallel processing without capacity limits has the property that the processing time does not depend on the amount of inputs. In other words, for a fixed time interval, its (mean) outputs are proportional to its (mean) inputs. The specified arrival pattern as a result of the preattentive stage has exactly this property: The probability of a stimulus coming out of the preattentive stage in the next infinitesimal time interval is proportional to the number of stimuli that have not yet finished this stage by the current time.

3.3.4.3 Number of servers, capacity and queue discipline

Regarding the other three characteristics, there are also many theoretical op- tions. Although empirical phenomena are usually very complex, it still appears advisable to start with models with the simplest options. Hence, I assume that the processing of stimuli by different processors is independent of each other (independent parallel service channels). The next question is how many parallel service channels are appropriate. Because the attentive stage involves the identification of visual objects, many theorists believe that the representations in this stage requires memory. Even parallel models assume a limited processing capacity. For example, TVA by Bundesen (1990) assumes that only four items at the most will be selected in the short term memory for further examination in parallel (final stage of the competition). However, clear empirical evidence on this issue is still lacking. Although a concrete number cannot be determined, it is very unlikely to be large (e.g., more than ten). The number of parallel servers is treated as a model parameter to be estimated, denoted by 𝑐. The number four may provide a good starting value.

The waiting room capacity concerns the question of whether there is a limit on the number of customers allowed in the system. Since the most theorists agree that the preattentive processing does not have a capacity limit, there appears to be no reason why any subsequently arriving stimuli will be discarded from the queue once a certain number of items are already in the system. Nevertheless, in the context of visual search, if there is a limit on the number of stimuli that have gone through the preattentive stage but not yet been attended to, then it is likely to be temporal fashion rather than quantitative. It seems more plausible

that such fragments of representation can co-exist in a large amount but the existence does not last long. That is, the customers in the queue may not remain in the queue after a certain amount of time. However, this kind of limit relies on the notion that the fragments of representation require memory to exist. But stimuli in a standard visual search task are constantly present before a response is made, which makes the memory of the physical features of the stimuli seemingly unnecessary. Therefore, the simpler assumption of unlimited waiting room capacity is adopted. For a given set size 𝑘, because there can be at most as many customers in the system as the population size , this is equivalent to the queue with a waiting room capacity of 𝑘.

As to the queue discipline, the question is whether to include prioritization. Prioritizing is plausible and part of some theories, e.g., GS (Wolfe, 1994a, 2007; Wolfe et al., 1989). Certain types of prioritization upon arrival can be equivalently modeled by prioritization of the incoming sequence, for example, the more similar an item to the target, the more likely it enters the queue in the first positions. I decide to reserve prioritizing as a theoretically convincing but more complex option and here assume a “first come, first served” rule (the next customer waiting in the queue will be assigned to any free server) instead.

In this way, the models are constrained to the kind of Markovian queueing model denoted by 𝑀/𝑀/𝑐/∞/𝑘/𝐹𝐶𝐹𝑆 (queueing model with exponentially distributed interarrival times and service times, 𝑐 servers, unlimited waiting room capacity, 𝑘 customers and default rule “first come, first served”).

3.4

Deriving RT predictions