4.3 Examining Single and Multiple Communications Circuits

4.3 Examining Single and Multiple Communications Circuits

One of the more common decisions facing a network designer is whether to use single or multiple communications circuits when connecting remote LANs via the use of bridges or routers. Obviously, from a reliability point of view, multiple circuits are preferable to the use of a single circuit. However, from a performance point of view, are two circuits, each with one half the transmission capacity of a single circuit, better than the use of a single transmission path ? This is the question that will be examined in this section, while leaving an investigation of the gain in availability and reliability of multiple circuits over single circuits for Chapter 6.

4.3.1 Comparing the Use of Single- and Dual-Port Equipment

Figure 4.9 illustrates the use of single- and dual-port remote routers or bridges to connect an Ethernet LAN to a distant Token Ring LAN. Suppose the single-port devices are connected to data service units (DSUs) that operate at 19,200 bps, whereas the multi-port device is connected to two DSUs, each operating at 9600 bps. Although each pair of communications devices has access to an aggregate transmission capacity of 19200, does the use of single and dual transmission paths in which the single path operates at twice the rate of each dual transmission path provide an equivalent level of service? To answer this question, let us return to the use of queuing theory. In doing so, let us assume that the number of transactions estimated to flow between LANs is 21,600 per eight- hour day. Let us also assume that the average frame size , including communications framing for transmission, is 1200 bytes.

Figure 4.9: Comparing the Use of Single- and Dual-Port (Multi-port) Networking Devices

4.3.2 Single Circuit

As a review of our prior discussion of Kendall's notation, we can consider the single circuit to operate on a first-in, first-out basis. When we assume that the number of frames can approach infinity, this means we can consider the single circuit communications link illustrated at the top of Figure 4.9a to more formally represent what is known as an M/M/1 queuing system.

4.3.2.1 Notation

The notation M/M/1 is actually a shorthand abbreviation of the form A/B/C, which is used by queuing theorists to describe queuing problems. The letter used in the A position represents the statistical characteristics of the arrival rate of items to be serviced, such as customers approaching a teller or frames flowing to a bridge or router. A probability density function (pdf), such as the Poisson pdf, is typically used to describe a customer arrival rate. One of the most commonly used pdfs is the exponential pdf, which is denoted as M in queuing theory shorthand notation. The exponential pdf is defined as:

This pdf is equivalent to saying that arriving customers behave as if they are not aware of each other's existence (i.e., occur randomly ). Thus, the arrival process is a process without memory and the shorthand abbreviation M used to denote an exponential arrival rate actually references the fact that the arrivals are "memory-less."

Readers should note that systems with an exponential inter-arrival distribution results in a Poisson arrival rate distribution. In fact, the probability that exactly n customers will arrive in a period of time t is given by the equation:

The above equation is the Poisson probability function. Thus, an exponential inter-arrival time distribution, which is memory-less and denoted in queuing shorthand notation by M, is generated by a Poisson process in which the arrival of any customer is independent of previous customers.

Returning to the A/B/C notation format used to describe a queuing problem, B represents the statistical characteristics of the server. A server that operates without regard to the length of a queue (e.g., does not get tired ) has its behavior described by an exponential service time distribution. Thus, such servers are also "memory-less" and the letter M is used in position B to describe the characteristics of our server. Finally, the letter C in the queuing problem shorthand description format represents the number of servers. Thus, the numeric 1 was used in our shorthand queuing abbreviation for position C to complete our description of the queuing model in Figure 4.9a.

4.3.2.2 Computations

Based on 21,600 transactions occurring in an eight-hour day, the arrival rate becomes:

Because it was assumed that the average frame size is 1200 bytes, the expected service time when the line operates at 19,200 bps becomes 1200 * 8/19200, or 0.5 seconds. Thus, the service rate becomes:

The utilization of the server (p) is the arrival rate divided by the service rate. Thus,

The probability that there are no frames in the system (P ) is one minus the utilization. Thus,

Let us continue and compute the mean number of frames expected in the system (L) and the mean length of the queue (Lq). Doing so, we obtain:

Thus, we can expect 0.6 frames to be in the system, while 0.225 frames, on the average, will be in the queue. Now let us focus attention on waiting times, the mean time waiting in the queue (Wq) and the mean time waiting in the system (W). Those two parameters are computed as follows :

Note that the difference between the average waiting time in the system of 0.8 seconds and the average waiting time of 0.3 seconds in the queue is 0.5 seconds. That time is exactly the expected service time for a 1200-byte frame to be carried on a 19,200-bps transmission circuit. Now that the basic queuing- related performance elements for the single-path circuit have been computed, let us focus attention on the use of dual transmission circuits.

4.3.3 Dual Circuits

The dual transmission path illustrated in the lower portion of Figure 4.9 represents an M/M/2 queuing system in which the numeric indicates there are two servers or, in our example, two channels or circuits.

Although you will normally think of a server as a device with memory that enables queues to form, you can also treat each circuit as a server. In doing so, each router or remote bridge then provides the buffer memory for the formation of queues for service or placement onto each communications circuit. Here, the use of two channels or communications circuits can be considered to represent a queuing system that is more formerly referred to as a multiple-channel, single-phase queuing system.

4.3.3.1 Computations

In computing the queuing parameters, note that the arrival rate of 0.75 frames per second remains the same. However, because each circuit now operates at 9600 bps, the expected service time is 1200 bytes * 8 bits/byte/9600 bytes, or 1 second. Thus, the service rate, ¼ , is 1/1, or 1 frame per second per transmission line.

For a multiple-channel, single-phase system, utilization is computed as follows where s is the number of servers.

So far, everything appears to be equivalent between a single transmission line and two lines in which each of the latter operate at one half the rate of the former.

In a multiple-channel, single-phase queuing system, the probability that there are no frames in the system (P ) is determined using the following formula:

where s represents the number of servers and n! and s! represent n and s factorial, where factorial n represents the value mx(m ˆ’ 1) * m ˆ’ 2) * * 1. The results of the computations for our two-server model to obtain the probability that there are no frames in the system are as follows:

This means that there is a 45 percent probability that both servers do not contain any frames at any point in time.

Now that P has been computed, we can use our prior computation in equations developed for multiple-channel, single-phase systems for other queuing parameters of interest. Those four parameters, Lq, L, W, and Wq are computed as follows:

4.3.4 General Observations

So far, we have made a number of computations. Now let us compare our previous computations and use the comparison between single and dual server models to make some general observations. Table 4.3 compares the queuing computations for the single and dual communications paths illustrated in Figure 4.9.

Table 4.3: Comparing Single- and Dual-Path Queuing Values

Parameter	Single Path	Dual Path
Arrival rate	0.75	0.75
Service rate	2	1
Utilization	0.375	0.375
Probability of no frames in the system	0.625	0.45
Mean number of frames in system	0.6	0.87
Mean length of queue	0.225	0.12
Mean time in system	0.3	1.162
Mean waiting time	0.8	0.162

In comparing the entries in Table 4.3 between the single and dual path scenarios, let us first focus attention on P , the probability that there is no frame in the system. Note that the single path has a higher probability that there is no frame in the system than the dual path. The reason for this can be explained by examining Figure 4.10, which indicates how transmission gaps can occur on two paths, each operating at a fraction of a single path.

Figure 4.10: Transmission Gaps Commonly Occur on Multiple Circuits

Suppose a 2400-byte frame arrives at a server connected to a single 19,200-bps data circuit. As indicated in the upper portion of Figure 4.10, one 19,200-bps circuit could transmit the first 2400-byte frame in precisely one second. If a second frame shows up precisely one second later, the first frame would have been transmitted and the transmission of the second frame can then commence.

If a second frame shows up just a fraction of a second after the first frame, the second frame will be queued for transmission. Then, once the first frame is transmitted, the second frame will immediately follow the first frame. In fact, the second frame can arrive at any time up to one second after the first frame without a transmission gap occurring. Now consider a similar situation in which two 2400-byte frames arrive one second apart at a bridge or router connected to two communications circuits. The first frame requires two seconds for transmission, while the second 9600-bps circuit is unoccupied for one second until the second frame arrives and is then transmitted on the second circuit because the first circuit is occupied. Because there is no way to split a frame between circuits, there is no way to fill the gap that occurred. The one-second gap illustrated at the lower portion of Figure 4.9 represents half the longest duration gap based on the previous assumptions. If a second frame arrived two seconds after the first, a two-second period of time would occur in which one circuit was not occupied. Similarly, if the second frame followed the first by 0.5 seconds, the gap would be reduced to a 0.5-second duration. As illustrated from this example, the use of two queuing systems, each with half the capacity of a single system will almost always have a lower level of performance due to the higher probability of occurrence of transmission gaps.

By comparing the single- and dual-path queuing values listed in Table 4.3, readers will note there are other significant differences between a single circuit operating at X bps and two circuits each operating at X/2 bps. For example, the mean number of frames in the system less the mean length of the queue tells us the mean number of frames on the circuit or circuits. Here, the use of a single path would result in 0.6 ˆ’ 0.225, or 0.375 frames, while the dual path would have 0.87 0.12, or 0.75 frames. This means that the mean number of frames in a two-circuit system is twice the number of frames found in a single-circuit system ” a situation we would intuitively expect.

Another area of difference between the single- and dual-circuit configurations concerns waiting times. Although the mean waiting time in the queue is less for a two-circuit configuration because it provides two paths for transmitting frames, its mean system time considerably exceeds that of a singlepath system. What this tells us is that the use of multiple circuits provides a larger "moving storage" facility for frames than a single-circuit system; however, the end result is that although queuing time (Wq) is reduced, the total time in the system (W) is increased. Thus, discounting availability considerations, a choice between a single circuit operating at X bps and two circuits each operating at X/2 bps should be made in favor of the single circuit.

4.3.5 Program QUEUE2.BAS

To facilitate computations associated with two-channel, single-phase queuing systems, the program QUEUE.BAS was modified. The results of that modification provide queuing statistics for a two-channel, single-phase queuing system for line rates ranging from 4800 bps to 1.536 Mbps. This program modification was renamed QUEUE2.BAS and its statements are listed in Table 4.4. In addition to revising the program to compute two-channel, single-phase queuing system statistics, the program was also modified to accept daily transactions, operational hours per day, and an average frame size as user input.

 REM PROGRAM QUEUE2.BAS CLS PRINT "PROGRAM QUEUE2.BAS - STATISTICS FOR TWO CHANNEL SINGLE PHASE QUEUE" PRINT REM AR=arrival rate REM MSR=mean service rate REM L=mean (expected) number of frames in system REM Lq=mean number of frames in queue REM W=mean time (s) in system REM Wq=mean waiting time (s) REM EST= expected service time INPUT "Enter transactions per day      : "; transactions INPUT "Enter operational hours per day : "; hrs INPUT "Enter average frame size        : "; frame AR = transactions/(hrs * 60 * 60) DATA 4800,9600,19200,56000,64000,128000,256000,384000,768000,1536000 FOR I = 1 TO 10 READ linespeed(I) est(I) = frame * 8/linespeed(I) MSR(I) = 1/est(I) UTILIZATION(I) = AR/(2 * MSR(I)) PROB0(I) = 1/(1 + (AR/MSR(I)) + (AR/MSR(I)) ^ 2/(2 * (1 - (AR/(2 * MSR(I)))))) Lq(I) = PROB0(I) * (AR/MSR(I)) ^ 2 * UTILIZATION(I) Lq(I) = Lq(I)/(2 * (1 - UTILIZATION(I)) ^ 2) L(I) = Lq(I) + AR/MSR(I) Wq(I) = Lq(I)/AR W(I) = Wq(I) + 1/MSR(I) NEXT I PRINT "Line Speed EST   MSR     Po       p        L        Lq W         Wq" FOR I = 1 TO 10 PRINT USING " #######  #.#### ###.## "; linespeed(I); est(I); MSR(I); PRINT USING " #.#####  #.#####"; PROB0(I); UTILIZATION(I); PRINT USING "  #.#####  #.#####  #.#####   #.#####"; L(I); Lq(I); W(I); Wq(I) NEXT I PRINT PRINT "where:" PRINT PRINT " EST= expected service time MSR = mean service rate" PRINT " Po=probability of zero frames in the system p = utilization" PRINT " L= mean number of frames in system Lq = mean number in queue" PRINT " W= mean waiting time in system Wq = mean waiting time in queue" 

Table 4.5 illustrates an example of the execution of QUEUE2.BAS. In this example, the program was executed using 21,600 transactions per eight-hour day, with an average frame size of 1200 bytes.

 PROGRAM QUEUE2.BAS - STATISTICS FOR TWO CHANNEL SINGLE PHASE QUEUE Enter transactions per day      : ? 21600 Enter operational hours per day : ? 8 Enter average frame size        : ? 1200 Line Speed EST   MSR     P   p       L       Lq       W     Wq    4800 2.0000   0.50 0.14286 0.75000 3.42857 1.92857 4.57143 2.57143    9600 1.0000   1.00 0.45455 0.37500 0.87273 0.12273 1.16364 0.16364   19200 0.5000   2.00 0.68421 0.18750 0.38866 0.01366 0.51822 0.01822   56000 0.1714   5.83 0.87919 0.06429 0.12910 0.00053 0.17214 0.00071   64000 0.1500   6.67 0.89349 0.05625 0.11286 0.00036 0.15048 0.00048  128000 0.0750  13.33 0.94529 0.02813 0.05629 0.00004 0.07506 0.00006  256000 0.0375  26.67 0.97227 0.01406 0.02813 0.00001 0.03751 0.00001  384000 0.0250  40.00 0.98142 0.00938 0.01875 0.00000 0.02500 0.00000  768000 0.0125  80.00 0.99067 0.00469 0.00938 0.00000 0.01250 0.00000 1536000 0.0063 160.00 0.99532 0.00234 0.00469 0.00000 0.00625 0.00000 where: EST = expected service time; MSR = mean service rate; P   = probability of zero frames in the system; p = utilization; L = mean number of frames in system; Lq = mean number in queue; W = mean waiting time in system; Wq = mean waiting time in queue. 

4.3.6 The Excel Model QUEUE2

To facilitate computations by those who prefer to use an electronic spread-sheet, another Microsoft Excel template was created. This template is stored in the file QUEUE2 in the directory Excel at the Web URL previously mentioned in this book.

Figure 4.11 illustrates the display of the QUEUE2 model for a frame rate of 21,600 frames per day, where the average frame length is 1200 bytes. Note that similar to the previous queuing model created in this chapter, this model displays eight minicharts that provide a visual indication of different queuing system parameters as the line speed varies.

Figure 4.11: The Excel QUEUE2 Model