QUEUEING NETWORKS AS MODELS OF HUMAN PERFORMANCE AND HUMANCOMPUTER INTERACTION Yili Liu Department of Industrial and Operations Engineering The University of Michigan Ann Arbor, MI 48109-2117 Technical Report 93-32 November 1993

Freprint To appear in Proceedings of the'94 Symposium on Human Interaction with Complex Systems Queueing Networks as Models of Human Performance and Human-Computer Interaction Yili Liu Dept. of Industrial and Operations Engineering University of Michigan 1205 Beal Ave. Ann Arbor, MI 48109-2117, USA ABSTRACT modeling human performance and human-computer interaction. This article describes several queueing network models of human performance and human-computer The idea of a queueing network arises interaction that we have developed recently and naturally when one thinks of a network of service illustrates the great value of queueing network stations (also called service centers, or simply, methods in establishing models of human nodes), each of which provides a service of some performance and human-computer interaction at all kind to the demanders for service (called customers), analysis levels and in establishing an integrated, either immediately or after a delay. Each center has a computational framework for unifying some currently waiting space for customers to wait if they cannot isolated models. The article starts with a theoretical immediately receive their requested service, and thus discussion of the most "micro" level of performance multiple queues may exist simultaneously in the and presents a queueing network model for reaction system. The service centers are connected by arcs time as a model of elementary mental processes. As a over which customers flow from node to node in the continuous-flow network model, it includes discrete network. Telephone communications systems, serial-stage and continuous-flow overlapping-stage computer networks and road traffic networks are models as well as discrete network models as special examples of queueing networks [1] [2]. cases. The second section moves to the more "macro", behavioral level and describes a 3-node It is not difficult to see the close queueing network model of multitask performance resemblance between a queueing network and the that includes single-channel, queueing theoretic current views of a human cognitive system. Before models and parallel-processing, multiple resources embarking on a detailed technical elaboration, the models as special cases. The third section reaches the following is a brief illustration of the potential level of applications and discusses queueing network relevance of queueing network concepts to human models of human-computer interaction and human- performance analysis. computer networks and their potential applications in standalone and networked environments. In order to In a queueing network representation of a illustrate the modeling capabilities of queueing cognitive system, the customers are stimulus networks, the first two sections discuss sojourn times components or information processing tasks, which and- waiting times in an open queueing network, and may enter the cognitive system at some node, traverse -the last section selects queue length distributions in a from node to node in the system, and depart from closed network as the tool of modeling, some node, not all tasks necessarily entering and leaving at the same nodes (e.g., the use of different sensory and motor modalities), or taking the same The computational models of human path once having entered the system (e.g., the use of performance and human-computer interaction that we separate memory and decision routines). Tasks may have developed recently and are described in this return to nodes previously visited (e.g., performance article are based on our view that human information feedback or decision loops), skip some nodes entirely processing system and human-computer systems are, (e.g., skill acquisition and automaticity), and even in many respects, analogous to a queueing network, remain in the system for a long time (e.g., memory in which information processing tasks may assume a rehearsal). Some customers may failed to enter a busy wide range of complex structural and temporal system (e.g., perceptual tunneling), leave a busy arrangements. The structural arrangements include system before they have been fully serviced (e.g., both serial selection and parallel execution, and the speed-stress induced errors), jockey for position by temporal arrangements include both immediate switching from one queue to another, or preempt activities and delayed processing. Queueing network earlier customers if the queue discipline allows this to methods employed widely in industrial engineering happen (task scheduling). Multiple queues may and systems analysis can serve as a valuable tool in improve their joint performance by adopting some 1

coordinated service schemes (e.g., task integration), For the most part of the paper, we will or lose effective communication in the face of mainly be concerned with queueing networks with overwhelming information (e.g., confusion, cross- the following characteristics: talk, and outcome conflict) [3] [4]. 1. Arrivals from the "outside" to node i follow a Poisson process with mean rate yi, In order to give mathematical substance to 2. Service times for each channel at the models presented in the following sections, we node i are independent and exponentially distributed introduce the following notations, which are now with parameter li, rather standard in the queueing network literature. 3. The routing probabilities (pij's) are Two sets of notations are needed, the first for independent of the state of the system, which is a describing a stochastic queueing process at a service vector representing the number of customers at each station, and the second for stochastic processes in a station. queueing network. Networks that have these properties'are A queueing process at a service station in a called separable networks or product-form networks. network is described by a series of symbols and They are also called Jackson networks, named after slashes such as A/B/C/D/E, where A indicates the the author who showed that this class of networks arrival pattern of customers as described by the have the following amazing property: the network probability distribution for interarrival-time or arrival acts as if each node can be viewed as an independent rate, B the probability distribution for service time, C M/M/c queue, with parameters Xi and gi. The joint the number of parallel service channels at the station, probability distribution for the number of customers D the restriction on waiting room capacity in front of at each node can be written as a product of marginal the station, and E the queue discipline (the manner by M/M/c's [5]. This amazing property makes it possible which the customers are selected from the queue for to derive many important results for the Jackson service). For the most part, this article will focus on network that are often not available or analytically the class of queueing process that has received most intractable for other types of networks. Jackson research attention and enjoyed a most fruitful history networks have subsequently received the most of producing usable analytical results. This queueing research attention and enjoyed a great success in process is denoted as M/M/c/oo/FCFS (or M/M/c for model development. The models have also been short), representing a queueing process with successfully applied in diverse areas, because exponential interarrival times (also called Poisson separable networks can be evaluated quite efficiently. arrival), exponential service times, c identical servers Furthermore, many authors have demonstrated that at a station, no restriction on the maximum number of many of the results for Jackson networks provide customers allowed in the queue, and first-come, first- close approximations to non-Jacksonian networks [6]. served queue discipline. The importance and In computer system analysis, the pragmatic, justifications of employing this type of queueing "operational" framework for queueing network process in performance modeling are discussed in all analysis, pioneered by Buzen and Denning, relies standard textbooks on queueing theory [2]. heavily on the assumption of separable queueing networks. It has been pointed out that, in practical We use the following symbols to represent a applications, inaccuracies resulting from violations of queueing network: Jackson's assumption typically are not worse than 1) K: number of nodes, those arising from other error sources (e.g., 2) i: identity of nodes, inadequate measurement data) [7] [8]. 3) yi: mean arrival rate to node i from outside the network (also called external arrival rate), With these notations and introductions in 4) pij: the probability that a customer visits hand, we are ready to present a number of queueing node j immediately after departing from node i (also network models for human performance and humancalled routing probability or switching probability), computer interaction. The presentation proceeds as i=l,..., K, j=O,.., K, with piO representing the follows. We will start with a theoretical treatment of probability that a customer leaves the network the most "micro" or "molecular" level of performance immediately after visiting node i, and present the models for elementary mental 5) Xt: the total mean arrival rate into node i processes, and then we move to the more "macro", (from outside and from other nodes), (according to "aggregated" behavioral level and describe a 3-node "traffic equation", Xi = yi + I pjiXj, summed over j=l model of multitask performance. The last section to K) reaches the level of applications and discusses 6) At: mean service rate for each channel of queueing network models of human-computer node i. interaction and human-computer networks and their potential applications in both standalone and 2

networked environments. In order to illustrate the additive factors method (1969) both assume nonmodeling capabilities of queueing networks, the first overlapping durations of serially arranged processes two sections focus the discussion on sojourn times [9] [10]. This class of models are also referred to as and waiting times in an open queueing network, and serial discrete-stage models. Donders assumed that the last section selects queue length distributions in a processes can be added or deleted from the closed network as the tool of modeling. processing chain while leaving intact the rest of the chain (called the assumption of pure insertion). Sternberg assumed that the duration of a process can QUEUEING NETWORKS AS MODELS OF be changed by experimental manipulations while this ELEMENTARY PSYCHOLOGICAL change will not produce indirect effects on the rest of PROCESSES the chain (called the assumption of selective influence). Based on the assumption of selective Why is there a delay between stimulus influence, Sternberg proposed an additive factors presentation and response initiation? This has been methodology for RT analysis, according to which one of the most enduring and fundamental questions experimental factors that influence a common process that psychologists have been fascinated with. will interact with each other, whereas those Although it appears that everyone can offer an influencing separate. processes will be additive. The answer to this seemingly simple question, the exact serial discrete-stage model and the additive factors nature of and the causes for this delay remain a methodology have been the fundamental basis of a mysterious domain. The current belief of cognitive large body of experimental literature. McGill and psychologists is that this delay is a reflection of the Gibbon (1965) noted that reaction time in a serial dynamic activities of an underlying mental discrete-stage model can be described by the generalarchitecture that transforms stimulus into response. gamma distribution, if the durations of each stage is And most importantly, since the cognitive system is exponentially distributed with different duration not amenable to open inspection, the characteristics means [11]. of this delay —also called reaction time (RT) —may offer important clues to the possible configurations of the mental architecture. Series Network Theoretical models that use reaction time as 1. the primary performance measure to infer the general rave structure of mental systems are also called models for Factors (Critical RT. Of great interest to the present discussion are two Discrete F rs Fac t ors3. General- Path) dimensions along which RT models can be classified. G a One of the two is a dynamics dimensionamma distinguishing discrete-processing from continuousflow models, and the other an architectural dimension distinguishing serial-stage models from network Continu- 1. Cascade 1.Queueing models. Discrete processing models assume that a ous- 2. Queue- Network mental process will not transmit its processing output Flow Series to other processes until it is completed and it transmits its output in an indivisible unit. Thus a process can not begin until all of its preceding processes are completed. Continuous-flow models Fig. 1. Reaction Time Models assume that each process transmits its available partial outputs to other processes continuously rather than waiting for the full completion of processing, Some theorists have studied sequentially and thus a process can begin processing even though arranged processes that permit temporal overlapping its preceding processes are still active. Serial-stage of process activities. Prominent among the RT models assume a serial arrangement, whereas models include McClelland's (1979) Cascade model network models assume a network configuration of [12], and more recently, Miller's (1993) queue-series the mental processes. The two dimensions jointly model [13]. Both models try to mimic the behavior of define four classes of models as shown in Figure 1. serial discrete-stage models, and examine the conditions under which the two classes of models In the top-left quadrant that defines serial converge or diverge in their predictions. This class of discrete processing models, we find the most models belong to the bottom-left quadrant of Figure traditional interpretation of information processing. 1. Donders' subtractive method (1868) and Sternberg's Townsend (1972) challenged the notion of 3

serial arrangement of mental processes and examined number, C, of distinct classes of components, with Ni possibility of parallel activities [14] Schweikert components of class i, i=l,..., C. In the simplest case, (1978) later developed a class of so-called PERT there is only one class of stimulus components that is networks [15], which assume that mental processes responsible to RT (this is the case considered by can be arranged as a network, with serial and parallel Miller). We may call them signal components. In a structures as special cases. Although the PERT more general case, there may be two classes of models allow processes that are not on the same path stimulus components — signal components and noise of a network to be active at the same time, they components. It will be shown below that this assume that processes on the same path operate in distinction is critical for queueing network analysis of strict sequence —a process can not start until all the some RT behavior such as speed-accuracy tradeoff preceding processes on the same path are completed. (SAT) and violations of the selective influence Furthermore, the PERT method for RT analysis assumption. It is easy to image situations in which a follows the postulate of selective influence and finer distinction between the classes of stimulus assumes that each experimental manipulation components is necessary, but this paper will not prolongs the duration of one process, but does not extend the discussion further to include those cases. change the duration of any other process. This class As nicely summarized in Miller (1993), "the stimulus of models can be classified as discrete-network components may be regarded as elementary stimulus models and is shown in the top-right quadrant of features, complex semantic codes, objects, or the Figure 1. associated neural activations" ([13], p.703), and as in Miller, this article will not attempt to develop the In the following, we present a queuing empirical means of identifying stimulus components. network model for elementary mental processes. The model, in its most general form, is a continuous-flow- Component arrivals and services. As network model. As will be shown below, the model pointed out by Pachella, the definition of stimulus takes the existing models in the other three quadrants onset is not always psychologically obvious [10]. of Figure 1 as special cases. Furthermore, this model Consistent with the common assumptions adopted in allows consideration of a broader range of possible queueing literature, the model assumes that the mental structures that can be subjected to empirical arrival sequence of stimulus components can be testing. The purpose is to expand the set of modeling described as a Poisson process. The model assumes tools available to psychologists and contribute to the that at the node i, customers have an exponentially psychological endeavor of discovering new mental distributed service time requirement with a mean of architectures as possible models of cognition. 1/ti. gi is often referred to as the service rate of node i. As discussed by numerous authors, this assumption General description of the model. is not as strong as it appears to be [2], and is also consistent with one of the most common assumptions General assumptions. The model assumes of other RT models [11] [14] [16]. that a reaction time task is carried out by a network of processing centers, each of which provides a distinct Reaction time as network sojourn time. type of information processing service to the From the perspective of RT analysis, the most customers (stimulus or task components). Analogous interesting performance measure of a queueing to other continuous-flow models, the model assumes network is customer sojourn time —the time a that each center can begin processing as soon as it customer spends in the network (or part of it). Several receives some customer (1 or more stimulus decades of queueing network research has shown that component) from outside the system or from another determining the sojourn time distribution of a center. A center immediately transmits any available customer in queueing networks is a very complicated output (a satisfied customer) to other centers or to the problem and among the hardest in queueing network outside of the network without waiting for the full theory. For non product-form networks, almost no completion of its processing of all its customers. explicit results exist. Even for product-form Similar to the cascade model and the queue-series networks, complicated correlations among waiting model [12][13], we assume that there is a separate times at various nodes exist and thus very little can be response unit at the end of the processing network, said about the sojourn times of a customer at which is activated when it has accumulated N signal successive nodes. An important exception to this components. statement is provided by the sojourn time distribution of a customer along a path in a product-form network Stim'ulus components as customers. I will when the path is "overtake-free", which means that adopt the term "stimulus components" used in Miller customers can not overtake or bypass one another. (1993) to refer to these customers or demanders [13]. When this overtake-free condition is satisfied, the The model assumes that a stimulus is composed of a sojourn times of a customer at various nodes along 4

the path are independent [1] [6] [17]. However, it (possibly infinite) of stimulus components should be emphasized here that this independence respectively. PERT networks are shown to be a does not mean that the sojourn times of successive special case of another type of queueing network customers are independent. As pointed out by Disney called acyclic fork-join networks when each stimulus and Konig (1985), the complete characterization of activates exactly one stimulus component. the joint sojourn times of a sequence of customers is still an unsolved problem ([1], p.377). Tandem Queues In this article, we only consider the types of Network sojourn time in a tandem queue. networks in which signal components cannot A special type of queueing network (also the simplest overtake each other, although noise components may type) is a tandem queue, also called a series queue, in overtake signal components (discussed below). For which the service stations form a series system with the models discussed in this article, we assume that flows always in a single direction from the first node each center has a single processing channel and a to the last node. Customers may enter from the FCFS discipline. The assumption of single channel outside only at node 1 and depart only from the last processing nodes is similar to that of Miller's queue- node. If each stimulus activates only 1 component, series model and is common in psychological theory. then the successive nodes that component has to visit will operate in strict sequence. In this case, we have a For this type of networks, the Nth signal serial discrete-stage processing system. If it activates component to depart from the network is also the Nth multiple components, successive nodes along the signal component to arrive from the outside. Thus, customer's route will operate with temporal overlap. RT is the sum of the Nth customer's network sojourn time (T) and the time interval between the first and More formally, we have a open network the Nth arrival (Ta). For Poisson arrivals, Ta follows where the Erlang distribution and is independent of T, which Y = X (i=l) means that Ta is neither influenced by nor offering= 0 (elsewhere) any insight into the structure of the network. Thus we and only need to analyze the network sojourn time of the pi = 1 (=i+l; 1 < i < k-l) Nth signal component (T). For Poisson arrivals, = 1 (i=k, j=O) signal components arrive at the network =0 (elsewhere) independently with each other, and thus the Nth It has been shown that the network sojourn signal component (called a tagged component) could time, T, for a M/M/1 tandem queue has as be any of the signal components with equal distribution the convolution [6] probability, and is stochastically indistinguishable Pr (T<t = (I- e(I -)t) * from any other signal components. * (- e4(PK)t), t>=O Let us use road traffic as an analogy. In which has been shown to be the general gamma order to study how changes in the traffic environment distribution: (e.g., road structure) influences the traveller Fk(t) = 1 - Cik e-(i-X))t (1) behaviror —in this case the time to reach a destination, where we can either study the travel time of a large number Cik = rl(gj-X)/(gj-k - (gi-k)) (2) of customers at once, or study a randomly selected Now let's compare this result with those of the serial "tagged" traveller a large number of times (who is discrete-stage and other continuous-flow models. either the same traveler or preferably randomly selected each time). We adopt the "tagged" customer Serial discrete-stage model. McGill and approach, because research results are only available Gibbon (1965) have shown that the general gamma is for this case. Actually, adopting this approach allows the RT distribution of a serial discrete stage model in us to model queuing networks that are not single- which the duration of each stage is exponentially channel FCFS-based. But we will not extend the distributed [11]. More specifically, McGill and present discussion into those cases. Gibbon showed that in a serial discrete-stage model with k stages, the RT distribution has the following With these important results in mind, we can form: proceed to analyze some interesting cases and Fk(t) = 1 - Cik e4(i)t (3) examine the previous models. We will show that the where discrete or cdntinuous-flow serial models are special 1/ti is the mean duration of the cases of a special type of queueing network called exponentially distributed passage time through stage tandem queues, corresponding to the situation in which each stimulus activates one or a large number 5

Comparing (1) and (3), it is clear that the one of the experimental factors affect the asymptotic general-gamma distribution of RT is not limited to level of activation. A particularly interesting result is serial discrete-stage models. Actually, the serial that the cascade model is able to fit the shape of the discrete stage model of McGill and Gibbon can be well-known time-accuracy curve closely. treated as a special case of the tandem queuing model by replacing (gji-3) with gi ((g.i-X) is often called the The readers may have already noticed the "effective service rate" of a node). The major general-gamma function in equation 4. In fact, a more conceptual difference is that the serial discrete stage general form of the tandem queueing model makes model has the largest possible grain size of the same set of predictions as the cascade model. transmission (a stimulus is indivisible). Only one Instead of allowing the Nth signal component to stimulus component is allowed in the tandem activate the response unit unconditionally when it network, and no other components are allowed to leaves the last node (always possessing enough enter the network until the current one has completed activation strength), the more general form of the processing (a situation in which k=0 and thus pi-k = tandem queueing model assumes that the responsegi). In the tandem queueing model, a stimulus is activation strength of the Nth signal component can regarded as consisting of a number of components, be manipulated by experimental factors. Analogous and they pass through the network like a traffic flow to the cascade model, we assume that in yes/no (k > 0). Furthermore, the tandem queueing model experiments, the response-activation strength of the allows the existence of noise components in the Nth signal component is ay/y, and the responsenetwork. Their main effect on RT is a reduction of activation strength of other stimulus components is effective service rate and the corresponding increase ay/n. As in the cascade model, we assume that actual in RT. In following analysis of the cascade model, we response execution is a discrete event that adds the will show that the introduction of noise components duration of a single discrete stage (e.g., 0.1 sec.) to facilitates modeling the well-known phenomenon of the time between the stimulus presentation and the speed-accuracy tradeoff. registration of the overt response. Then for the tandem queueing model, the observed value of d' at McClelland's Cascade model. McClelland time t is given by proposed a cascade model as a continuous-flow d'(t) = (ay/y- ay/n) Fn[t-.1]/ l+a2(rn[tserial-processing model [12]. The model assumes that.1])2 1/2 (5) the human information processing system functions This equation is identical to equation (13) of [12], like a series of parallel linear integrators. These linear which has been shown to fit the time-accuracy curve integrators take a weighted sum of a subset of theclosely outputs of the integrators at the preceding level and produces continuous output that is always availableThe derivation of equation should be the for processing at the next level. The central same as that for equation (13) of [12]. The major assumption of the cascade model is that the rate of difference between the two continuous-flow models activation of a linear integrator depends on the is in the interpretation of the general-gamma difference between the level its output are driving it function. In the cascade model, the general gamma to and the level of activation the unit has alreadyfunction, ), is an activation function and function, Fn(t), is an activation function and reached..A cascade equationri —the heart of the cascade r ed. ascade equation-the heart of te cascarepresents the relative activation of a unit at Level n model —was derived based on this assumption, which at time t. In the tandem queueing model, the same gives an expression for the activation of linear.. u t e function represents the probability that a beingintegrator j at processing level n to a stimulus S pre a tim t = 0. T e io h t observed stimulus component has passed through the presented at time t = 0. The equation has the following form r ime t ^ O.last node of the network adn thus reached the following form: followg f,,r.m: response unit at time t. anj/S(t) = anj/S(1 - iKi e-(ki)t) (4) where anj/S is the asymptotic activation of the linear Miller's queue series model. Miller's model integrator that would result if the stimulus were left considered a special type of tandem queue [13]. In on indefinitely, and the kis are the rate constants of Miller's model, stimulus components arrive at the the different processes in the system. McClelland queue series at the same time (called bulk arrival in examined the effects of manipulating rate constants queueing literature). Components are not served in and asymptotic levels on RT and derived a set of the same order as they arrive at the various servers prediction of RT behavior. Similar to the serial (not FCFS). Therefore, the nth customer to depart discrete-stage models, the cascade model shows that from the end of the queue series is not likely to be the experimental factors affecting the rate of the same nth to arrive at the front of series. Miller used PERT process will interact, whereas those affecting the rates representation and numerical simulation to examine of different processes are additive. However, the the time for N customers to traverse through the predictions become more complicated when at least system and concluded that, within the class of queue 6

series models he considered, experimental factors affecting different processing stages always have Acyclic Fork-Join networks additive effects on reaction time with sequential stages but rarely do so with overlapping stages, and This is a special type of queueing network, thus, observations of factor additivity support in which a "fork" node simultaneously creates several discrete-stage models. As Miller stated: "From the new customers, which are sent to separate queues. ubiquity of additive factor effects in RT experiments, and the corresponding join occurs at a "join" node it appears that nondiscrete queue-series models must when the services of all these new customers are be regarded as fairly implausible general descriptions completed. Apparently, just as the serial discreteof human information processing". ([13], p.712). stage model can be treated as a special case of the tandem queueing model, a PERT network can be As mentioned above, virtually no analytical treated as a special case of an acyclic fork-join result is available in the queueing literature about network, in which each stimulus activates only one sojourn times in the type of tandem queue considered stimulus component (N=I). No other stimulus in Miller's model. It appears that Miller's conclusion components are allowed to enter the network until the should only be restricted to the type of nondiscrete offsprings of the newly admitted stimulus component model he considered, since the nondiscrete M/M/1 have completed processing. tandem queue we considered and discussed above mimics McGill and Gibbon's discrete serial model Stochastic PERT networks and fork-join precisely. There are at least two related issues that are queueing networks are both extremely difficult to worthy of further exploration. First, in Miller's analyze. Schweikert considered deterministic PERT simulation, the time for the ith component to pass networks [14]. Fisher and Goldstein considered through stage j (tij's) were either constants or stochastic PERT networks using a method they independently randomly selected from each of three proposed called order-of-processing (OP) diagram distributional families: normal, uniform, or [16]. Fork-join networks is a new research area in the exponential-plus-constant. However, as mentioned queueing network research community [18]. Future above, the sojourn times of a customer at successive breakthroughs in the research on stochastic PERT non-FCFS stages are dependent random variables, networks or fork-join queueing networks will rather than constants or independent random hopefully improve the applicability of these methods variables. Second, the queue-series model follows the for RT analysis. postulate of selective influence —an essential assumption for discrete models. "In the queue-series Simon-Foley Queueing Network model, a factor affecting stage i changes only the values of tij" ([13] p.711.). There has been a We have mentioned that if a Jackson substantial amount of debate about how reasonable network does not allow customers overtake each this assumption is for discrete models. For a other (e.g., if single-channel Jackson network has continuous-flow model, this assumption should be atmost one path from node i to node j for every i, j) even more debatable. then sojourn times at nodes are mutually independent and exponentially distributed random variables and Miller has pointed out that although the passage time along a path can be described as a queue-series model is able to approximate the shape general gamma distribution, which has been shown to of the activation functions of the cascade model, the play a central role in the McGill and Gibbon's model, two models produce different effects on RT. The the cascade model, and the tandem queue model. We explanation was that the cascade model allows now consider what happens if a network allows experimental factors to have downstream effects, customers to overtake or bypass each other. whereas the queueing series model does not consider such propagation. This could also explain why the A classic example of a non-overtake-free predictions of the cascade model and the tandem network is the so-called Simon-Foley network queueing models converge, while both diverge from (Figure 2). Simon and Foley (1979) considered a the queue-series model. Both models allow the three-node Jackson network with single servers at effects of experimental manipulations to propagate each node [17]. Customers only enter the system at through the system, and neither model assumes node one and exit the system at node three. After selective influence. Without a further clarification of visiting node a customer goes directly to node 3 these issues, the following statement should be with probability (1-p), or goes to node 2 with regarded as tentative and debatable: "In view of the probability p. If the customer goes to node 2, he goes prevalence of additivity, then, the most plausible directly to node 3 after receiving service at node 2. conclusion is that nondiscrete queue-series models Simon and Foley showed that the sojourn time in the are generally inappropriate" ([13], p.713). first and the third queue (T1 and T3) are not 7

independent for those customers who go through the Although previous studies have discussed the second queue. T1 and T2 are independent, T2 and T3 possible existence of indirect influence of are independent, T1 and T3 are independent for those experimental factors [9] [10], Simon-Foley network customers who go directly from node 1 to node 3. offers a possible method of testing and quantifying Foley and Kiessler later showed that T3 is one possible type of indirect influence. Furthermore, stochastically increasing in T1 for a customer that since Simon-Foley network is among the simplest goes through node 2, i.e., P{T3 > tiTl} is increasing continuous-flow network that is neither strictly serial in T1. A result that is particularly useful for mean nor parallel, data collected will provide critical sojourn time analysis is derived by Walrand and insights into the architecture and function of human Varaiya [19], who showed that cognitive system. E{T31TI=t'} >E{T31TI=t}, t'>t>0 (6) where E{T} represents the mean of T. In this section, queueing network methods are applied to the analysis of reaction time and elementary mental processes. Customers in a network 1-p _ are indistinguishable components of the same ~~~~00_-^^~~~ - ---— ~ ^stimulus or of the same task. The models and the methods can be extended to situations in which Pn^~~~~ ^ ___ customers in a network are components of separate P _AL....^_~ ~stimuli or parts of separate tasks. The next section partly illustrate this point through the models of multitask performance. Fig.2. A Simon-Foley Network (The network does not follow Sternberg's assumption of selective influence) QUEUEING NETWORKS AS MODELS OF MULTITASK PERFORMANCE These results are important in that they may This section applies the queueing network suggest a new class of mental architectures that can methods to the analysis of human multitask be subjected to empirical tests. The test for the performance. We are concerned with psychological existence of a Simon-Foley arrangement of the behavior at a more macro and aggregated level than in psychological processes makes no more assumptions the previous section. The modeling work has a strong than those for testing the validity of the Schweikert's motive for application, and the approach is engineering PERT methodology for RT analysis. It assumes that and approximative. A 3-node queueing network model we are able to prolong the duration of a process of is described in this section that integrates the concerns interest, and we are able to record time at several of single channel, queueing theoretic models of points in the network [15]. According to Equation 6, selective attention and parallel processing, multiple if in a task situation in which prolonging a process resources model of divided attention. The two schools produces a corresponding increase in the duration of of models have fundamental differences in their views another process, then there is a great possibility that of the nature of multitask performance and in their the task situation involves a Simon-Foley network of research and modeling methodology. The single mental processes, particularly if such a network also " channel, serial processing models treat multitask'makes sense' in terms of other knowledge" [9]. For performance as an issue of task selection and example, it is possible that in certain task situations scheduling. The multiple resources, parallel processing node 1 has the function of distinguishing signal from models, in contrast, treat multitask performance as an noise components. After passing through node 1, issue of parallel allocation and division of processing signal components must go through node 2 for a high resources among simultaneous tasks. level cognitive analysis, while noise components go directly from node 1 to node 3. Experimental From the perspective of computational manipulations that change T1 or p will produce modeling, the single channel assumptions have thus far corresponding changes in T3 that are positively enjoyed a greater success, as indicated by the existence correlated with changes in T1. of a set of well-established models such as the queueing theoretic models reviewed in the following section. We are in the process of reviewing These models provide formal mechanisms for published data in the literature to search for possible representing and codifying the single channel evidence of' this mental network. A series of assumptions of task selection in computational terms. experiments are also being prepared to test this The multiple resources models, in contrast, have only method. Results along this line of investigation recently started to see some of their concerns being should have significant theoretical implications, gradually accommodated in several simulation models 8

of human performance, and there is still a lack of a set experiment that simulated a multitask flight of computational methods to transform the assumptions management situation [27]. A similar task scenario was of simultaneous execution and resource allocation into also used in an earlier study by Walden and Rouse engineering terms. Furthermore, there does not exist a (1978) that investigated the suitability of a single server set of integrated engineering-based methods to model queueing model of pilot decision making [28]. the concerns of both schools of models and to bridge Greenstein and Rouse (1982) integrated a pattern the gap between the two. As indicated in a recent report recognition technique called discriminant analysis with of the Committee on Human Factors of the National queueing theory methods in their 2-stage model of Research Council report, "there is no unique method to human decision making in multi-process monitoring model the two most important features of macromodel: situations [29]. task selection and simultaneous execution ([20], p.40)." Although the single channel based models After a brief review of the queueing theoretic and the have demonstrated tremendous success in modeling multiple resources models, we will illustrate that visual sampling and task scheduling, "We must queueing networks may provide a useful method for recognize that people in fact can do more than one thing modeling the two features. at a time and normally do (Adams, Tenney and Pew, 1991; [30], p.5)". In multitask situations, single channel Queueing theoretic models of selective attention assumptions and the analogy of a single-CPU timeshared computer or a single server queueing system These models postulates that the human may only capture part of the nature of human functions like a time-shared computer with a single performance and may not be adequate to portray the central processing unit (CPU), which quickly switches complex cognitive mechanisms for concurrent and allocates its processing capacity among a variety of processing. It may be necessary to address the parallel tasks in a sequential and all-or-none fashion. The processing aspect of performance, to consider the models view human multitask performance as a single intensity as well as the time characteristics of task server queueing problem or multitask sequencing demand, and to analyze the structural similarity of problem in which multiple tasks or diverse sources of concurrent tasks. These issues have been the focus of information are queued for service from the human investigation of models of divided attention. information processing system [21]. Multiple Resources Models of Divided Attention A number of queueing theoretic models have In contrast to the single channel assumption been developed, focusing on human visual sampling that attention capacity can only be switched in a and monitoring behavior. Senders (1966) developed a sequential and all-or-none fashion among competing instrument monitoring model, which integrated the tasks, divided attention theorists have generally single channel concept and the sampling theorem of suggested that the limited human information Shannon's information theory in making its predictions processing capacity can be simultaneously allocated to about the observer's fractional dwell time on each multiple tasks in a graded fashion, and that information monitored instrument [22]. Carbonell (1966) proposed a for simultaneous tasks can be processed in parallel as single server priority queueing model of multi- long as the total processing load does not exceed a instrument visual sampling and used simulation to solve person's processing capacity. Since the 1970s, the the model [23]. Senders and Posner (1976) further concept of capacity has evolved and become more developed the queuing theoretic approach to instrument commonly known as a "resource". This concept has monitoring and provided analytical solutions to a model also evolved from referring to a single undifferentiated that they developed for display sampling [24]. Schmidt pool to models that human information processing (1978) applied queueing theoretic method to the should include multiple pools of resources (Wickens, analysis of air traffic control task [25]. 1984; [31]). The models suggest that task interference should only be manifest to the extent that they compete An extensive effort in applying the queuing for the same pool of resource. Of the various definitions theoretic methods to the modeling of human machine of processing resources that have been proposed so far, systems can be found in a series of studies conducted by the one that has received the most consensus in the Rouse and his colleagues. Rouse (1977) described literature is the definition based on a distinction human-computer interaction as a queueing system with between spatial and verbal processing codes. The the human and the computer as two servers [26]. He dichotomy of spatial and verbal processing codes formulated a queueing theoretic model of dynamic distinguishes operations of perception, working allocation of responsibility between the human and the memory and response that have a linguistic and computer in multitask situations, and illustrated the symbolic base from those that have a spatial analog potential utility of this model with simulation base. Ample evidence has demonstrated the role of experiments. Chu and Rouse (1979) later investigated processing codes in accounting for variances in task the predictive power of the model with a behavioral interference. Recently, several simulation models of 9

human performance, e.g. the MICROSAINT model spatial task but could still perform a secondary verbal [32] and the WINDEX model [33], have started to tasks such as a light conversation. But as the difficulty accommodate some of the assumptions of the multiple of driving further increases —such as driving on a resources models. winding and narrow road in a stormy weather or in a Since the multiple resources models were heavy-traffic area —the driver would have to concentrate originally proposed to address the characteristics of on driving, which could be easily disturbed by a slight parallel allocation of scarce yet divisible processing disruption of any kind. resources to concurrent activities, the models do not Since the queueing theoretic models have not provide a formal mechanism to model the serial attempted to address the parallel and structural aspects processing bottlenecks and the selective and scheduling of task demand, it is not clear how the models would aspects of task performance. Thus, a typical research account for differential effects of spatial and verbal strategy of investigators in this research paradigm has tasks. A natural approach to this problem from the been to treat these bottlenecks of serial processing as queueing theoretic perspective might be to develop a set extraneous factors or to keep the influence of these of complicated scheduling algorithms that allow the factors as small or constant as possible. However, as single server to vary task priorities and service rates reviewed in the previous section, processing bottlenecks according to the processing codes of the to-bedo exist and play an important role in many task processed tasks. But it is not clear whether this is a situations. The next section will argue that the existence feasible approach, and the problem remains until the of processing bottlenecks might be a major cause of algorithms are developed. A concept that might be interference between tasks that do not use the same invoked by the multiple resource theorists in explaining processing code. the existence of between-codes interference is the concept of concurrence cost. But the concept appears A 3-Node Oueueing Network Model of Multitask limited in this context, since it fails to explain why Performance between-codes interference could be absent in some A variety of experimental studies have situations, but present in other situations, and perhaps demonstrated that task interference is expected to be more importantly, why between-codes interference greater when concurrent tasks use the same processing could show a pattern of gradual increase as seen in a code than when they require separate codes. However, performance/difficulty tradeoff, and why the the data do not indicate that two tasks demanding concurrence costs exist. separate codes will always be perfectly time-shared While it remains to be seen how the single [31]. The data seem to suggest a pattern of task channel and the multiple resources models would interference as follows [3]. First, task interference will address these issues, the following three-node queueing not be observed when the total demand of the network model offers a plausible account of the concurrent tasks is low, regardless of the processing research findings. This queuing network model has a codes involved. Second, when the total task demand is parallel processing component and a serial processing sufficiently high, both within-code and between-codes component. The parallel processing component refers to interferences could be observed, but within-code the parallel operation of a "spatial server" (S) and a interference is more likely to occur. Third, increases in "verbal server" (V), analogous to the spatial and verbal task difficulty would produce a faster increase in processing mechanisms or resources advocated in the within-code than between-codes interference. multiple resources theory. The serial processing It appears that this pattern of task interference component refers to the serial operation from either S or is also consistent with intuition and experiences. For V to the third server, which can be tentatively referred example, in a perfect driving environment an easy to as the central server (C). secondary task can be performed concurrently with the primary driving task without causing any performance decrement, no matter whether the easy secondary task 00 is spatial (e.g., imagining a simple road map or tuning a radio) or verbal (e.g., recollecting a previous cmE conversation or talking to a passenger). But under the. _ same driving condition, a difficult secondary spatial task (such as mentally "walking" around a complex spatial layout or using a complicated navigational Figure 3: A 3-Node Queueing Network Model of device) would be more likely to disrupt driving than Concurrent Spatial and Verbal Tasks would an equally difficult verbal task (such as reciting a difficult poem or engaging in a challenging conversation). As the driving environment becomes The model assumes that spatial and verbal more hostile, a driver would very likely experience tasks take separate routes of the network: a spatial task great difficulty in performing even a simple secondary must be serviced by the server S, and a verbal task must 10

be serviced by the server V, before they receive the the difficulty of concurrent tasks increases. Waiting required service of the server C prior to their time approaches infinity as the arrival rate approaches completion. The capacity of the servers in meeting the the service rate, indicating a situation in which the tasks service needs of the arriving tasks can be represented as are too difficult to be performed simultaneously the service rate of the servers (p). (the number of (beyond the processing limit). Therefore, of most customers that can be serviced per unit of time). The interest to the model is the change in performance when model also assumes that a task is composed of a number X is smaller than g at each server. of task components, and the arrival rate (X) (the number As seen in Figure 3, customer waiting could of arriving customers per unit of time) of the task occur in front of S or V or both servers (similar to the components is a rough measure of the difficulty or the predictions of the multiple resources models), or in service demand of a task. As in other queueing theoretic front of C (similar to the predictions of the single models of human performance, this model assumes that channel or queueing theoretic models), or a there is a performance cost associated with delaying combination of both cases. A way to model the pattern service to a task (called the cost of waiting). of task interference discussed earlier in this section is to assume that the serial processing bottleneck (server C) Using the symbols introduced at the beginning has a capacity that is smaller than the sum of the of the article, the essential constituents of the 3-node capacities of the spatial and the verbal server, but queueing network can be represented as follows: greater than the capacity of either of the two servers. 1) K=3, That is, 4LC < iS + pV, pC > pS, and pC > pV. For 2) i=l, 2, 3, representing the spatial, verbal, example, if we assume that pC=14, iS=10, pV=10, and the central server, respectively, where iC, iS, pV refer to the capacity of the three 3) yi =i, for i = 1, 2 server nodes respectively, then the relationships =0, for i = 3 between waiting time and arrival rate at the three 4)Xi = Xi, for i = 1,2 servers are shown in Figure 4 and Figure 5. If we = XI +X2, for i=3 further assume that the cost of waiting in front of the 5) P12 = P21 = P31 = P32 = 0, servers are identical, then the relationship between pi = 0, for Vi, performance decrements and task difficulty should P13= P31 =1 show a similar pattern as Figure 2 and Figure 3. 6) PlO = P20 = 0, P30=1 Different patterns of relationship can be obtained if we make different assumptions about the cost of waiting. A number of performance measures can be Four important cases of customer waiting (task computed using queueing network methods. Of most interference) emerge in the 3-node network: interest to the present analysis is the customer waiting Case 1: This is the case when the customer time in front of each server. We continue to assume that arrival rate at each of the centers is significantly smaller the network is a separable queueing network, and for than the service rate (i.e., Xi < pi) (a situation in which this type of network, we have, easy tasks are time-shared with each other). Minimum Wqi = 1/(p.i-Xi)- 1/i, waiting is expected, and thus performance decrements where Wqi is the mean waiting time of customers in would be minimal in performing the simultaneous front of server i, tasks, regardless of whether the same or separate i is the mean service rate of server i processing codes are involved. Xi is the total mean arrival rate at server i.Case 2: When the customer arrival rate approaches the service rate of either the spatial (S) or the verbal server (V) but not that of the central server, For separable queueing networks, the arrival significant waiting is expected in front of S or V, rate at the central server is the sum of the arrival rates at respectively, but not in front of C (e.g., when Xs=8, the spatial and the verbal servers. By making Xv=l, Xc=9). This is the case when within-code assumptions about the capacities of each server and the interference is the only source of task interference. This cost of waiting in front of each server, the queueing within-code interference will increase quickly as the network model allows the modeling of a variety of arrival rate to the congested server S or V continues to patterns of task interference, considering both the increase. difficulty and the processing codes of concurrent tasks. Case 3: When the customer arrival rate We assume that the pi's are constants, which is an approaches the service rate of the central server but not assumption consistent with one of the most common that of the spatial or the verbal server, customers are assumptions in multitask research. The relationship expected to wait in front of C, but not in front of S or V betwen iting t rat ara n (e.g., when s=6,waiting time and arrival rate is diagrammed inis the case when Figures 4 and 5. It can be seen that waiting time between-codes interference is the only source of task increases monotonically as the arrival rate increases, interference. Progressive increase in Xs or v will indicating that performance decrements will increase as produce progressive increase in Xc, and produce a 11

corresponding progressive increase in the betweencodes interference. However, as long as Xs and Xv are both still much smaller than gs and gv, within-code interference would be minimal. QUEUEING NETWORKS AS MODELS OF Case 4: This is the most general case, in which HUMAN-COMPUTER INTERACTION AND performance decrements are caused by a combination HUMAN-COMPUTER NETWORKS of within- and between-codes interferences (a combination of Case 2 and Case 3). Due to the heavy In essence, models of multitask performance demands of the tasks, queues could be observed in front reviewed in the previous section are also models of of all three servers. human-computer interaction, although the role of the computer is not explicitly addressed. In this section, 4 ___________we consider the joint function of the human and the computer agents in a human-computer system. The two types of agents could very well be in charge of different functions and have different performance 3 _ features, and we use queueing networks with different types of servers as a general modeling 0 framework, which apparently treats the queueing s4 theoretic models of a single server or identical servers E 2- _ * as special cases. 2 Along this line of thinking and modeling H' philosophy, we have developed and are currently. j validating several queueing network models of - _.human-computer interaction, both in standalone and 1; in networked environments. Since the previous two.*;~~~ I ~sections of the paper have demonstrated the value of ~'sojourn times and waiting times as performance 0.. I-..' " measures, this section shifts the attention to another 0 5 10 15 20 important stochastic process and performance measure —queue length distributions (or the states of a Arrival Rate queueing network system). Also, since networks in the previous two sections are both open networks, we Figure 4: Relationship between waiting time turn to closed networks in this section. However, it and arrival rate at the spatial or verbal server should be obvious that sojourn time, waiting time and open networks can be similarly applied in modeling 4 human-computer interaction as well. In a closed network, the same customers circulate eternally through the network. A closed network can also be interpreted as an open network 3 _.- with the total number of customers held fixed. In this system, a new customer arrives when and only when SQ)Eg~~~~~~ ~~~a customer leaves the system. The sliding window -4.protocol of message communication and the paging policy in computer memory management are 0 * examples of this type of system. -rfi"^~~~~~~~~ ~~~One of the models we have developed is in *~.~~~~~^4~~~ ~ ~the context of a failure management system (e.g., an BS^~~~ 1_I~~~~ -aircraft or a process plant). In the simplest, standalone situation, a human and a computer work together to detect and correct system failures. One or more of the system components could fail (e.g., one 0.... "' or more engines or one or more boilers). The system 0 5 10 15 20 is designed in such as way that when a system component fails, the computer will work on it first Arrival Rate (e.g., to perform detection, warning and preliminary analysis functions) and then the human controller will Figure 5: Relationship between waiting time work on it to perform higher level tasks. After the and arrival rate at the central server human controller have successfully completed his/her 12

task, the failed component will return to normal computer network. A specific example is a failure operation. This human-computer-machine system can management system in which there are two type of be modeled as the simple queueing network model machine component failures, each type is handled by shown in Figure 6. For the 3-node (machine, a computer and then by a human operator. A possible computer, human) closed-series network, the state of scenario is that two computers and two human the queueing system at any time instant t is a vector operators work cooperatively in a manner illustrated p(nl, n2, n3), representing the number of customers in Figure 7, where the "copilot" completes his/her at node ni (i=l, 2, 3) at time t. For the system task alone with a probability of p, but need to forward described above, p(nl, n2, n3) means that there are the problem to the "pilot" with a probability of (1-p), nl machine components in normal operation, n2 before the component is returned for normal being serviced by the computer, and n3 by the operation. human. The total number of customers (denoted by S) In order to compute the queue length should be a known quantity (e.g., S=k could mean distributions, we need the routing probability of the that there is a total of k engines). customers —pij, the probability that a machine component will immediately visit node j after departing from node i, which is specified by the task structure. In Figure 7, we have,;2 1~ A ~p12 = q (the probability that a failure is of type 1), p13 = 1-q (the probability that a failure is of type 2), p54 = p (the probability that human operator Machine Computer Human 2 needs help from human operator 1), p56 =1 - p (the probability that human Fig.6. A closed queueing network model of human- operator 2 can complete his/her job alone) computer interaction in a failure management system p24 =p35 =p46 p60 =pOl =1 described in the text pij = 0, for all other i and j's. The expected value of the number of appearances of node i on a routing is computed with the following recursive equation, p(nl, n2, n3) can be computed easily with ei = pOi + Jim=1 (em Pmi) the following set of equations, derived from the With a total of k machine components in the results of Jackson (1963) [34]: queueing system, p(nl, n2, n3, n4, n5) can be p(nl, n2, n31S=k) = co*(nl, n2, n3)/T*(S=k); computed easily with the following set of equations, co*(nl, n2, n3) = 13i=l 1ki1=(1/Clij); derived from the results of Jackson (1963): T*(S=k) = *(nl, n2, n3), summed over p(nl, n2, n3, n4, n5IS=k) = co*(nl, n2, (nl, n2, n3) with S=k; n3)/T*(S=k); co*(nl, n2, n3, n4, n5) where gij is the mean service rate of node i when 15si=lHki=l(ei/gij); there are j customers at node i. Apparently, the T*(S=k) = Yco*(nl, n2, n3, n4, n5), summed "service rate" of the machine (node 1) is the rate at over (nl, n2, n3, n4, n5) with S=k; which it causes machine components to fail. The values for uij are usually obtainable from A number of question can be answered with measurements, specifications or historical data. the computed queue length distributional values. The The above set of equations allow us to type of questions include the relative workload of predict a number of interesting performance features operator 1 versus operator 2, the proportion of time of the system. For example, it is easy to compute the during which the machine has at least c components proportion of time during which the human operator operating normally, and the effects of changing will have at least one machine component to repair network configuration or service rates. (2p(nl, n2, n3), summed over (nl, n2, n3) with n3>0 and S=k), or the proportion of time during which the Although the models are presented in the machine will have at least two components working context of a network of human and computer agents normally (e.g., at least two engines are running) interacting with each other toward a common goal, an (Xp(n 1, n2, n3), summed over (nl, n2, n3) with n l> area known as computer-supported cooperative work and S=k). (CSCW). The same methodology can be applied to We have extended the work to modeling the broader area of human-computer networks, which more complicated systems involving more than one also includes situations in which competitive or humans and more than one computers —a human- confrontive agents may compete with each other for 13

limited network resources and cause delays in servicing other agents' processing needs. [2] L. Kleinrock, "Oueueing Systems," New York: Wiley, 1975. [3] Y. Liu, "Visual scanning, memory scanning, Computer 1 Human 1 and computational human performance modeling," Proc. of the Human Factors Society 37th Annual Meeting, 1993. _4~-~ ( > >) IT > [4] Y. Liu, "A queueing network model of human ---- sq X^ __P multi-task performance." Tech. Rep. Univ. of Michigan, Machine j I Dept. of IOE. 1993. [5] J. Jackson, "Networks of waiting lines," Computer 2 Human 2 Oper. Res., vol-5, pp.518-521, 1957. Fig. 7. A queueing network model of a human- [6] 0. Boxma and H. Daduna, "Sojourn times in computer network in the failure management system queueing networks," In H. Takagi (Ed.), Stochastic described in the text Analysis of Computer and Communications Systems, p.401-450, 1990, North Holland. Although a multitude of human-computer,Although a multitude of human-compuer [7] J. Buzen, "Fundamental operational laws of networking tools and CSCW applications have been computer system performance Acta Informatica computer system performance," Acta Informatica, developed, there is a substantial lack of predictive 167-18, models and theories. As Schneiderman (1992) vol-7pp.67-121976. pointed out, this is a "vast uncharted territory: P D., theories are sparse, measurement is informal, data a s of ng neok ode oputin analysis is overwhelming, and predictive models are nonexistent" ([35], p.391). The model presented in Surveys, vol-10, pp.225-261, 1978. this section illustrates that queueing network methods could serve as a useful tool for establishing [9] S. Sternberg, "The discovery of processing performance theories and predictive models of stages: Extensions of Donders's method," Acta human-computer networks and for establishing Psychologica, 30, p.276-235, 1969. theory-guided, systematic ways of performance measurement and analysis, particularly the issues of [10] R. Pachella, "The interpretation of reaction concern involve timing, scheduling and resource time in information processing research," In B. allocation. Kantowitz (Ed.), Human information processing: The models presented in Figures 6 and 7 are Tutorials in Performance and Cognition, p.41-82, currently being evaluated with lab experiments using 1974, Hillsdale, N.J.: Erlbaum. a simulated failure management system and human subjects. We are also in the process of preparing [11] W. McGill and J. Gibbon, "The generalexperiments to validate a model of human-computer gamma distribution and reaction times," J. of Math. network with competing agents. Psvch., 2, 1-18, 1965. We hope that this article has illustrated the [12] J. McClelland, "On the time relations of potential power of queueing network methods in mental processes: An examination of systems of establishing new models of human cognition, human processes in cascade," Psychological Review, 86, performance and human-computer interaction on 287-330. various analysis levels, and in establishing an integrated, computational framework for unifying [13] J. Miller, "A queue-series model for reaction some currently isolated models. time, with discrete-stage and continuous-flow models as special cases," Psychological Review, 100, 702REFERENCES 715, 1993 [1] R. Disney and D. Konig, "Queueing [14] J. Townsend and F. Ashby, "The Stochastic [l ] R. Disney and D. Konig, "Queueing networks: A survey of their random processes," Modeling of Elementary Psychological Processes." SIAM Review, vol-27, 335-403, 1985. Cambridge: Cambridge Univ. Press, 1983. 14

[15] R. Schweikert, "A critical path decision making responsibility between human and generalization of the additive factor methods: computer in multi-tsk situations," IEEE Trans. SMC-9, Analysis of a Stroop task," J. of Math. Psych., 18, pp.769-778, 1979. pp. 105-139, 1978. [28] R. Walden and W. Rouse, "a queueing model [16] D. Fisher and W. Goldstein, "Stochastic of pilot decisionmaking in a multitask flight PERT networks as models of cognition: Derivation of management situation," IEEE Trans. SMC-8, pp.867the mean, variance, and distribution of reaction time 875, 1978. using Order-of-Processing (OP) diagrams," J. of Math. Psvch., vol-27, 121-151, 1983. [29] J. Greenstein and W. Rouse, "A model of human deciison making in multiple process monitoring [17] B. Simon and R. Foley, "Some results on situations," IEEE Trans. SMC-12, pp.182-193, 1982. sojourn times in acyclic Jackson networks," Management Science, vol-25, 1027-1034, 1979. [30] M. Adams, et al., "Strategic workload and the cognitive management of advanced multi-task [18] F. Baccelli, et al., "Acyclic fork-join systems," CSERIAC SOAR Report 91-6, 1991. queueing networks," J. ACM, vol-36, pp.615-642, 1989. [31] C. Wickens, "Processing resources in attention," In Parasuraman, R. and D. Davis (Eds.) [19] J. Walrand and P. Varaiya, "Sojourn times Varieties of attention. New York: Academic Press, and the overtaking condition in Jacksonian 1984. networks," Adv. Appl. Prob., vol-12, 1000-1018, 1980. [32] K. Laughery, "Micro SAINT: A tool for modeling human performance in systems," in G. [20] S. Baron, et al., (Eds.) "Quantitative Modeling McMillan et al., (Eds.), "Applications of Human of Human Performance in Complex. Dynamic Performance Models to System Design." pp.219-230, Systems," Washington, DC.: National Academy Press, New York: Plenum, 1989. 1990. [33] R. North and V. Riley, "W/INDEX: A [21] K. Pattipati, and D. Kleinman, "A review of predictive model of operator workload," in G. the engineering models of information-processing and McMillan et al., (Eds.), "Applications of Human decision-making in multi-task supervisory control," In Performance Models to System Design," pp.81-89, Damos, D. (Ed.), Multiple task performance. pp. 35-68, New York: Plenum, 1989. 1992. [34] J. Jackson, "Jobshop-like queueing [22] J. Senders, "The human operator as a monitor systems," Management Sci., vol-10, pp. 131-142, and controller of multidegree freedom systems," IEEE 1963. Trans. on HFE. HFE-5, 2-5, 1964. [35] B. Schneiderman, "Designing the User [23] J. Carbonell, "A queueing model of many- Interface," 2nd Ed., New York: Addison-Wesley, instrument visual sampling," IEEE Trans. HFE-4, 1992. pp.157-164, 1966. [24] J. Senders and M. Posner, "A queueing model of monitoring and supervisory behavior," in T. Sheridan and G. Johannsen (Eds.), Monitoring Behavior and Supervisory Control. New York: Plenum, 1976. [25] D. Schmidt, "A queueing analysis of the air traffic controller's workload," IEEE Trans. SMC-8, pp.492-293, 1978. [26] W. Rouse, "Human-computer interaction in multitask situations," IEEE Trans. on SMC-7, 384-391, 1977. [27] Y. Chu and W. Rouse, "Adaptive allocation of 15