8.5. Queues, Waiting Time, and Response Time
During its execution, a transaction in e-commerce systems is serviced by many different resources, such as Web servers, database servers, and payment servers. Each time a transaction or request visits a resource, it may need to queue for the use of the resource. Figure 8.10(a) shows the graphical notation used to represent a resource (a circle) and its queue (striped rectangle). The resource in the case of Fig. 8.10(a) could be an authentication server, and the striped rectangle would represent the queue of transactions waiting to be authenticated by the server. From Fig. 8.10-a, we notice that
Figure 8.10. Graphical Representations of a Resource and Its Queue.
Equation 8.5.15
where response time is the time spent by an e-commerce transaction per visit to the resource, queuing and receiving service.
In some cases, there may be multiple resources for the same queue. Consider the example of an e-commerce site with multiple servers and some sort of load balancing mechanism, used to distribute the requests among the servers. In this case, there is a common queue of requests waiting to be scheduled to any of the servers, as shown in Fig. 8.10(b).
There are situations in which a resource is dedicated to a request or an ample number of resources exist so that no queuing takes place. We call these resources delay resources since they only impose a delay to the flow of a request. The graphical representation of a delay resource is a circle without the striped rectangle (the queue). Delay resources can be used to represent the time needed by a third party (e.g., payment authorization or ad generation) to service a request from an e-commerce site.
As the load on a resource increases, more requests will be queued for the resource. At light loads, however, the total queuing time may be negligible when compared with the service time at the resource. In these cases, we may want to represent the resource as a delay resource to simplify the model. We now define the notation used throughout this book to represent performance variables for queues in e-commerce systems. Some of the concepts presented here were already introduced, albeit more informally, in the previous sections. We will call queue the waiting queue plus the resource or resources associated with the waiting queue. Let us use the following notation.
Vi is the average number of visits to queue i by an e-commerce transaction.
Si is the average service time of an e-commerce transaction at resource i per visit to the resource.
Wi is the average waiting time of an e-commerce transaction at queue i per visit to the queue.
Ri is the average response time of an e-commerce transaction at queue i, defined as the sum of the average waiting time plus average service time per visit to the queue. Therefore,
Equation 8.5.16
λi is the average arrival rate of requests to queue i.
Xi is the average throughput of queue i, defined as the average number of transactions that complete from queue i per unit time. We will assume that any queue is observed during a large period such that the number of arrivals and departures to the queue are almost the same. This assumption is known as Flow Equilibrium Assumption [7] [13] and implies that λi = Xi.
x0 is the average system throughput, defined as the average number of transactions that complete per unit time.
is the average number of transactions waiting at queue i.
is the average number of transactions receiving service at any of the resources of queue i. In the case of a single resource queue (see Fig. 8.10-a), 
is a number between zero and one that can be interpreted as the fraction of time that the resource is busy, or in other words, the utilization of the resource.Ni is the average number of transactions at queue i waiting or receiving service from any resource at queue i. Therefore,
Equation 8.5.17
