Chapter 4: Understanding and Applying Waiting Line Analysis

Overview

One of the more interesting areas of mathematics that many readers experience every day is waiting line analysis, also commonly referred to as queuing theory. Every time we approach a toll plaza , enter a fast food restaurant, and walk to position ourselves into a line or line up in a bank for service, we are in a waiting line. Over the past 30 years , a considerable amount of mathematical research has been performed that resulted in the development of a series of queuing systems that can be used to model different types of waiting lines. Fortunately for network personnel, this effort can also be applied to a variety of communication systems. For example, consider the operation of remote bridges and routers.

One of the major problems associated with the use of remote bridges and routers is the delay those devices introduce into linked networks. This delay results from the fact that a local area network (LAN) operating rate is normally an order of magnitude or higher than the wide area network (WAN) operating rate used to connect LANs via the use of remote bridges or routers. A second major problem associated with the use of remote bridges and routers concerns the selection of an appropriate amount of buffer memory for installation in each device. If too little memory is installed in such devices, the probability that memory will be filled when a frame arrives for forwarding increases . This, in turn , results in the inability of a bridge or router to service the frame, causing the originating network station to periodically regenerate the frame or a timeout to occur, which results in the termination of an existing communications session. Thus, an insufficient amount of buffer memory can result in both extended network delays as well as an increase in network traffic. The above problems are also applicable in modified form to LAN switches. If too much buffer memory is configured for switch ports, delays in the traversal of data could cause such real-time applications as voice and video to provide unacceptable results. In comparison, an insufficient amount of buffer memory would cause frames to be dropped. Because real-time voice transported over IP applications will not retransmit dropped packets, when too many are dropped, the receiving parties may think they are communicating with Bugs Bunny. Thus, packet loss and packet delay can be significant problems for VoIP (voice-over-IP) applications.

Through the use of queuing theory we can develop models to determine the delays associated with the use of remote bridges, switches, and routers. In addition, we can investigate the effect of modifying the operating rate of communications circuits used to interconnect those devices. Doing so will enable us to examine the effect of different communications circuit operating rates upon equipment delays, as well as to understand that beyond some operating rates, further increases in the operating rate of a communications circuit will have an insignificant effect on equipment and network performance.

Once we obtain an appreciation for the use of queuing theory to determine an acceptable operating rate for WAN transmission facilities linking LANs, we will examine a related problem. That problem involves the use of queuing theory to determine whether to use single or multiple communications circuits when connecting LANs with respect to their ability to service frames arriving at a remote bridge or router connected to single or multiple circuits. In examining this problem we will expand our knowledge of queuing theory to cover multiple channel, single-phase queuing systems, as such systems represent the flow of data through a remote bridge or router connected to multiple communications circuits. In concluding this chapter we turn our attention to the use of queuing theory to determine the minimum amount of buffer storage remote bridges and routers should contain to provide a predefined level of performance. In doing so we will develop a six-step approach readers can follow to select equipment with a sufficient amount of buffer storage to satisfy their specific organizational networking requirements.