Modeling Overloaded Voice over Internet Protocol Systems

Modeling Overloaded Voice overInternet Protocol SystemsMartin J. Fischer, Ph.D.Denise M. Bevilacqua Masi, Ph.D. Since September 11, many telecommunications systems have seen the need to deal with overloading during high traffic conditions. As the use of Voice over Internet Protocol (VoIP) increases, an important question needs to be answered—“Can the voice packets be delivered in a timely fashion when there has been a signifi-cant increase in traffic?” In this paper, we considered the problem of modeling VoIP systems in an over-loaded condition. We looked at the problem from a simulation and analytic point of view. We presented analytic models for the packet latency, jitter, and loss probability for two prevalent disciplines being used for VoIP today: First Come First Served and Priority Queueing. A third queueing discipline, Weighted Fair Queueing, is also discussed. In addition, we investigated how simulation languages like GPSS/H compared with respect to runtime with simulation models developed using Microsoft Visual Basic for Applications and if their runtimes are acceptable for practical use.BackgroundThe events of September 11 have signaled the poten-tial for telecommunications systems to see significant periodic increases in traffic loading for extended peri-ods of time. As the use of Voice over Internet Proto-col (VoIP) increases, a question needs to be ad-dressed—“What effect would the increases in traffic have on the performance of VoIP systems?” There can be up to a 400 percent increase in the normal traf-fic load a network experiences during a crisis like Sep-tember 11. [1] This increase can have a significant impact on the National Security/Emergency Prepared-ness (NS/EP) community and their ability to commu-nicate during this highly critical time.This paper has grown out of our modeling and simulation support to the National Communications System (NCS). The mission of NCS includes assisting the president, National Security Council, Office of Science and Technology Policy, and the Office of Management and Budget in the coordination of the planning for and provision of NS/EP communications for the federal government under all circumstances, including crisis or emergency, attack, recovery, and reconstitution. In support of its mission, NCS runs several emergency telecommunications priority ser-vice programs for federal government users, including the Government Emergency Telecommunications Ser-vice (GETS) and the Wireless Priority Service (WPS). However, industry is moving from circuit switched to IP technology for all telecommunications applications including voice, and NCS is initiating a new program called the Next Generation Priority Service (NGPS). This NCS program will investigate the need to evolve toward IP capability to ensure continuity of priority traffic during emergencies. As part of that investiga-tion, we are conducting various modeling and simula-tion activities to support the NGPS. The results pre-sented in this paper are just one step in gaining a better understanding of the modeling of such systems.In this paper, we examine the availability of mod-els, both simulation and analytic, to address the performance of VoIP systems under heavily loaded conditions. Simulation models are investigated with respect to runtimes, and whether or not home grown models need to be developed instead of using com-mercial off-the-shelf (COTS) products. Analytic mod-els are investigated with respect to availability and accuracy when approximations are used.The three main performance measures for voice and data are packet latency, jitter, and loss. At the Applied Telecommunications Symposium in 2005, [2] we discussed the available analytic results for the three major queueing disciplines in use today:First Come First Served (FCFS)Priority Queueing (PQ)Weighted Fair Queueing (WFQ)Also discussed at the symposium were the appropriate statistical distributions that could be used to model the voice and data packet arrival process as well as their packet sizes. These distributions are the ones used in this paper.In an earlier paper submission, [3] we concen-trated on the situation where the total load (voice plus data) was less than one and the buffer sizes were large.In this paper, we consider the case where the buffer sizes are finite and the total load is greater than one. We present analytic results for the three performance measures and two major queueing disciplines, FCFS and PQ. Some of these results are exact and some are approximate. The approximations are compared with simulation results and those are presented here. The analytic algorithms are presented next; and then we present our numerical investigations from these ana-lytics. A final analysis on packet arrival distributions is given later, and then concluding remarks are pre-sented.FCFS, PQ, and WFQ Disciplines AnalyticsLet the voice and data packet arrival be independent Poisson processes with rates λv and λd ; a later section in this paper and information from our 2005 sympo-sium [2] discusses the validity of this assumption. We assume the data packet size distribution is D 3 and given as in Table 1. (See reference [3] for discussion as to how this distribution was generated.) Based on the codec that is used, the voice packet size, D, will be fixed. Let S d and S v be the random variables repre-senting the length of time to transmit a data or voice packet. It will be a function of the packet size and line speed.The data load is }{d S E d d λρ= and similarlythe voice load is }{v S E v v λρ=. We are considering a queueing system with two classes of customers, where the arrival processes for each class are independent Poisson processes and the data packet service time is D 3 and the voice D. In what follows, we concentrate on the situation where the total load, d v ρρρ+=, isgreater or equal to 1. The case of ρ < 1 was consid-ered in another paper. [3] For the overload case, we must have finite buffers for both types of traffic. The three measures of per-formance for both voice and data which are most im-portant are latency, (E{W v } and E{W d }), jitter (Jit{W v } and Jit{W d }), and loss probability (PL v and PL d ).Internet ProtocolPacket Size (bits)Percentage 320 604,600 1612,000 24Table 1. Data Size Packet DistributionFCFS DisciplineUnder the FCFS discipline, there is one buffer for both types of traffic; denoted by its size as K. Packets are placed in the queue in the order of arrival and served in the same order. Since the sum of Poisson processes is Poisson, the queueing system is an M/D 3+D/1/K+1 system where the arrival rate is d v λλλ+=. (The shorthand M/D 3+D/1/K+1 uses Kendall’s notation for describing queueing processes. [4] It represents a queue with Poisson arrivals, discrete service times, a single server, and a buffer size of K.) Let p i , i = 1, 2, 3, 4 be the probability of an i size packet. Without loss of generality, we have data packets with p i = pc i d λ/ λ for i = 1, 2, 3, where pc i is given in Table 1. The voice packets are denoted by I = 4 and p 4 = v λ/λ.Let q j be the probability an arrival sees j packets in the system. For the M/D 3+D/1/K+1 queue, we have for j = 1, 2, … K [5]∑=−+−=ji i j a i q j a q j q 110λλ (1)and for j=K+1∑−=−−=+11)1(01K k i q q K q ρρ. (2)The quantity q 0 is found from the fact that .110=∑+=K j j qThe factor a j is given bydt j j t t e t B j a ∫∞−−=0!)())(1(λλ (3)where B(t) is the service time probability distribution and (1-B(t)) = 1 for t < S 1, (1-B(t)) = 1-p 1 for S 1 < t < S 2, (1-B(t)) = (1-p 1-p 2) for S 2 < t < S 3, (1-B(t)) = 1- p 1-p 2-p 3 for S 3 < t < S 3 and (1-B(t)) = 0 for t > S4. HereS i is the length of time it takes to transmit an i th -sized packet. An important point is that equations (1), (2), and (3) also hold for the case where ρ <1. Once the q j are found the expected number ofpackets in the buffer as seen by an arrival, E{Q q }, isgiven by ∑+=−=11)1(}{K j j q j q Q E . The probability apacket is rejected is given by PB v = PB d = q K+1. IfE{W q } is the expected length of time a packet spendsin the buffer, then by Little’s formula we have))1(1(}{}{+−=K q q Q E q W E λ. (4)The voice and data packets spend the same length of time in the buffer and so the respective time in the system is E{W v } = E{W q } +E{S v } and E{W d } = E{W q } +E{S d } for voice and data.The final quantity of interest, jitter, is equal to the standard deviation of the system delay for voice and data. It is the square root of the variance of the system delay, which is the sum of the variance of the bufferwaiting time plus the variance of the service time. The variance of the buffer waiting time is the same for both voice and data and can be approximated as fol-lows.Let Var{W q } be the buffer waiting time variance, we have:2}{}{}{}{}{S E Q Var S Var Q E q W Var +=(5)where S is the random variable representing the ser-vice time of an arbitrary packet and Q is the random variable representing the total number of packets in the system, and we have Pr{Q = j} = q j . This devel-opment starts with approximating the queue waiting time random variable W q ~ Q*S. That is, a packet’s queue waiting time equals the number of packets in the system it sees upon arrival, times the packet ser-vice time. This is not exact as the time to complete service for the packet being serviced upon his arrival is not the same as a packet that just started service; thus, equation (5) is not exact. Using W q = Q*S and results from H. C. Tijms, [5] one can get an expression for }{Wq Var given by equation (5).The Jit{W v ) for voice is just the square root of thesum of the variance of the queue waiting time and thevariance of the voice packet service time. But thevariance of the voice packet service time is zero soJit{W v ) =(Var{W q })0.5. Similarly, we have Jit{W d } =(Var{W q }+Var{S d })0.5. Later in this paper, we exam-ine the effectiveness of the approximation }{Wq Var .PQ Discipline Under the PQ discipline, there are two buffers, one for the voice packets and the other for data, having buffer sizes K v and K d , respectively. Each time the server becomes free, it transmits the voice packets waiting in the buffer before any data packets are transmitted. There is no interruption of service, so if a data packet is in service and a voice packet arrives to an empty voice buffer, it waits until that data packet is served and then is serviced.For the case where d v ρρρ+=< 1, and thebuffers are large, expressions for the voice and data packet latency and jitter are presented by J. W. Cohen.[6] Here we assume ρ > 1. If the voice load is greater than 1 (ρv > 1), the voice packet delay can be modeled as an overloaded M/D/1/K v +1 system and the results[5] given above and specialized to M/D/1/K v + 1 hold. The data packets never gets access to the server and so their performance measures are E{W d } = Jit {W d } = 0 with PB d = 1.The more interesting case is when ρ > 1 and ρv < 1. For this case, when the number of voice pack-ets in the system is zero, there are data packets waiting to be served and they are transmitted until voice pack-ets arrive. So, the first voice packet to arrive will have to wait until the data packet finishes service before itis served. These types of queueing systems are known as queues with vacations or breakdowns. [6] For the voice traffic behavior with ρv < 1 and a fi-nite buffer, we follow the method for M/D/1/K v+1. [5, 6] This method involves using a Markov Chain analy-sis to generate the state probabilities for the M/D/1 queue, and then truncate and normalize them to ac-count for the finite buffer. The latency and packet loss equations are exact, but the jitter results are obtained as was done in equation (5). We modify the equations for generating the probability of n voice packets in an infinite buffer M/D/1 queue to account for the excep-tional first service. That is, we modified the Markov Chain analysis equations to account for the fact that when no voice packets are present, the first service to a voice packet is composed of a regular voice service plus the residual service of the data packet. The resid-ual service time of the data packet is given in manyplaces. [5] This analysis is not completely correct as it assumes that when one or more voice packets are inthe system the voice packet are receiving service. Wecould have one or more voice packets in the systemand the data packets still in service. The accuracy ofthis approximation will be considered in the next sec-tion.For the data traffic with ρv < 1, we approximatethe performance for data by partitioning the totalbandwidth into the amount that is available to data and not being used by the voice. Using the method de-scribed above, we know how much of the bandwidth is being used by the voice traffic, ρv (1-PB v ). Since the bandwidth is being used 100 percent (with ρ >1) of the time, the bandwidth available to data is (1- ρv (1-PB v )) times the speed of the line. Using this as the band-width available to data, we use the analysis presented in reference [5] for M/D 3/1/K d +1, where the data ser-vice time is adjusted to account for the slower speedline. For the expected queue delay for data, we useequation (4) where the state probabilities were gener-ated via the method just described. The expected ser-vice time for data would be computed using the full bandwidth. The variance of the buffer delay for data is determined using equation (5), but where the service distribution has been modified for the slower rate. The variance of the service which is to be added to the buffer delay variance is found using the service time distribution with the full rate. Thus, we approximate the buffer delay for data by only using the line speed not being used by voice. For the actual data service time, we use the full line speed. The numerical exam-ples in the next section examine how good this ap-proximation is.A point to make is that if ρ > 1, then the line is busy 100 percent of the time and the carried total load of both voice plus data traffic is 1. Thus, when we compute the behavior for voice, the blocking for data can be found from the total carried load being equal to 1.WFQ DisciplineUnder the WFQ discipline, there are two buffers—one for voice and one for data—and the bandwidth is di-vided and allocated to each class of traffic. That is, so much bandwidth is given to voice and the rest to data. Since the voice packets are performance sensitive and the voice packets arrival rate and size is known, the amount of bandwidth allocated to voice is usually enough to ensure ρv < 1. In any event, once the band-width has been allocated, the voice performance can be found from M/D/1/K v+1 queue results, and the data performance is found from M/D3/1/K d+1, as long as ρv < 1 and ρd < 1 and no overflowing is allowed. Over-flowing allows one class of traffic to use the band-width from another class that is not being used. If overflowing between classes of traffic is allowed ac-cording to pre-set weighting parameters, the analysis gets much more complicated and is an open area for research.Numerical ExamplesIn this section we look at some numerical examples and simulation results for the FCFS and PQ disci-plines. Analysis methodologies for the WFQ disci-pline is a future research area.We compare using GPSS/H to Microsoft Visual Basic for Applications (MS VBA) to write the simula-tion model. Because we are dealing with overloaded systems, runtimes may become prohibitive and writing specialized code may reduce these times. We also compare the analytic results with those of the simula-tion. The data packet sizes were shown earlier in Ta-ble 1, and the voice packets are being generated using a G.711 codec with size equal to 1.608 kb.FCFS Discipline ExamplesWe look at two types of links—an OC-48 link (i.e., transmission speed of 2396 kb per ms) and a T3 link (transmission speed of 44.736 kb per ms). For the T3 and OC-48 link examples, we held the voice and data loads to be 0.4228 and 0.9299, respectively, with a total load of 1.3527. We compare the GPSS/H and MS VBA simulation results with respect to runtimes, and analytic results with respect to accuracy. In gen-eral, it is not recommended to write simulations in a general-purpose, non-simulation programming lan-guage due to longer model development times and challenges in extending such models; [7] however, in the case of a simple single-server queue, with a large number of packets, one may wish to consider a gen-eral-purpose, non-simulation programming language like MS VBA.For the OC-48 example, the GPSS/H and the MS VBA simulations were run for five million packet ar-rivals. The MS VBA simulation had three warm-up periods—0, 50000, and 100000—before statistics were collected. The first two warm-up periods were used in the GPSS/H runs. In this example, the buffer size was 9690 slots.There are three main points to be drawn from this example (as shown in Table 2). First, GPSS/H has a significantly longer run time than the MS VBA simu-lations, but both are quite long. The reason is the MS VBA code can be streamlined to collect and generate the desired outputs. GPSS/H is a general-purpose simulation language and has a significant amount of overhead. This is magnified when trying to simulate overloaded queues with large buffer sizes.The second point is this—the loss probability and expected system delay results are not significantly impacted by the warm-up period, as is the jitter. The system delay and jitter are given in ms. We observe that since the expected service time and variance for voice and data packets is small, there is minimal dif-ference between the voice and data performance measures.Finally, we expected the system delay and block-ing given by the analytics and the simulation to agree, but we knew the analytic expression for the jitter was not exact. For this example, we see that it is quite good and, therefore, we would expect the jitter ap-proximation to be good for overloaded systems. The point to be made is that if one is simulating these sys-tems, and jitter is a performance measure of interest, then one should build a warm-up period into the simu-lations.Warm-Up PeriodsPB E{Wv} E{Wd} Jit{Wv} Jit{Wd}Time(Seconds)0.259 10.74 10.75 0.553 0.553 3557 0.261 10.79 10.79 0.139 0.139 3590 MS VBA 0 50000 1000000.261 10.79 10.79 0.139 0.139 3599 0.260 10.75 10.75 0.568 0.570 19392 GPSS/H500000.261 10.79 10.79 0.149 0.149 19512 Analytic0.261 10.79 10.79 0.145 0.145 31Table 2. OC-48 Link ComparisonsThe first example established that GPSS/H took longer than the MS VBA simulation and that a warm-up period was required to obtain accurate statistics for the jitter. The first observation makes sense, as is shown in Table 3, however, to discover what happens with respect to the second observation when the line speed is reduced to a T3. For that example, the voice and data loads were the same as in Table 2 and the buffer size was 606.For this example, we see that the warm-up period does not have a significant effect on the jitter calcula-tion. In addition, our analytic approximation for jitter is again quite good. Also, the runtimes for the GPSS/H were significantly faster than MS VBA, a flip-flop from the first case. We attribute that result to the fact that buffer size is smaller (606 compared to 9690 for the OC-48 link). As the number of packets in the model increases, the GPSS/H memory and over-head requirements become much greater as compared to our MS VBA model.Our experiments have shown that either GPSS/H or MS VBA can be successfully used to generate a simulation model for an overloaded FCFS VoIP sys-tem. The simulation runtimes are significantly longer than the analytics and may not be acceptable.The analytics are accurate and require signifi-cantly less time to run. These differences are magni-fied for overloaded systems and the need to reach steady state.The final point we would like to make is this—the results given in both tables for the loss probability, PB, are the same and equal to 0.26. We know that the voice and data loads for both examples were identical, but the buffer sizes are significantly different, 9690 compared to 606. So why did we get the same loss probability? The reason stems from the well known result [5] that for ρ > 1, as the buffer gets large, then2607.3527.13527.)1(==−→ρρPB . The system can carry one Erlang of traffic and the remaining traffic islost. So, as long as the buffer size is large, one Erlang of traffic will be carried and the rest will be lost.PQ Discipline ExamplesIn this section, we compare the analytic results with those on a GPSS/H simulation for the PQ discipline. We had several analytic approximations for the de-sired measures of performance and we need to see if they were acceptable. For this comparison, we con-sider a T3 line with a voice buffer size of 300 and a data buffer size of 1000. Figures 1, 2, and 3 show the voice packet latency, blocking probability, and jitter as a function of voice load when the data packet load is 0.25.We approximated the voice packet performance by using results for M/D/1/K v +1, [5, 6] except we modified the equations for generating the probability of n voice packets in an infinite buffer M/D/1 queue with exceptional first service. We see that the ap-proximation is good for the probability of blocking and latency.Next we look at the data packet latency, probabil-ity of blocking, and jitter for the same parameters as we did for voice measures of performance (see Figures 4, 5, and 6).To generate these measures of performance when ρ > = 1 and ρv < 1, we use the link bandwidth avail-able to data as (1- ρv (1-PB v )) times the speed of the line and use the analysis in reference [5] for M/D 3/1/K d +1, where the data service time is adjusted to account for the slower speed line. For ρ < 1, the results given in reference [6] are used for data.For data, the analytics are approximate for all measures of performance. They get worse when com-pared to simulation results as the voice load grows. As the voice load approaches 1, data packets are re-ceiving poorer and poorer service until they do not receive service at all. The analytics and simulation show this behavior and track each other. In general, for both cases, we see our results are acceptable, butWarm-Up Periods PB E{Wv} E{Wd} Jit{Wv}Jit{Wd}Time(Seconds)0.260 35.92 35.97 1.99 1.99 2330.5 0.260 35.93 35.98 1.94 1.94 1943.4MS VBA50000100000 0.260 35.94 35.98 1.94 1.94 1835.10.261 35.95 36.00 2.03 2.03 473.40.261 35.97 36.02 1.97 1.97 478.0GPSS/H50000100000 0.261 35.97 36.02 1.97 1.97 482.3Analytic 0.261 35.98 36.03 1.95 1.95 0.1 Table 3. T3 Link ComparisonsFigure 1. Voice Packet LatencyFigure 2. Voice Packet BlockingFigure 3. Voice Packet JitterFigure 4. Data Packet LatencyFigure 5. Data Packet Probability of BlockingFigure 6. Data Packet Jitterget worse as the voice load grows to around 1. Al-though not as good as the voice analytics, the data analytics are acceptable.For the case of ρ < 1, we used an infinite buffer model. For the finite buffer case, that begs the ques-tion—when does the blocking become a factor as a function of the load? That is, as ρ approaches 1 from below, what happens to the blocking probability? To examine this behavior, we use the FCFS model shown earlier, with the data load fixed at 0.25 and the voice load increasing. The results shown in Figure 7 were generated using equation (2), [5] above.We see that except for small buffer sizes, the total load must be greater than 0.98 before the blocking is measurable. Therefore, assume using the infinite buffer model with ρ < 1 is good except when the total load is very close to 1.Packet Arrival Distribution AnalysesWe have assumed in the previous sections that the packet arrival processes for data and voice are Pois-son. This section further investigates the appropriate-ness of that assumption. When individual loads are less than 70 percent, we have previously seen that a Poisson process approximation is quite good. [2, 3] As these loads individually approach 1 and get larger, this approximation may not be valid. Here we investi-gate the validity of this assumption for the voice traf-fic. It is more sensitive than the data traffic. We con-sider the situation as the voice load gets large and use the methods discussed in other documentation by the authors. [2, 3]For the three main performance measures (la-tency, packet blocking, and jitter), Figures 8, 9, and 10 compare the results of a simulation for the Sriram and Whitt voice packet arrival process [8] and the results of using an M/D/1/K model [5] for a T1 line, a G.711 codec, and a buffer size of 120. For this example, when the number of voice calls equals 53 (the voice load is 1), we see that the expected buffer delay and packet blocking using Poisson packet arrival agree quite closely to the simulation, whereas the packet jitter agrees within a large area, but seems to be worse when the load is in a region near 1 (number of voice calls equals 53).In Figures 11, 12, and 13, we extend the compari-son conducted for T1 to the case of a T3 line with a buffer size of 300. Again, we see similar results as those for the T1 line. The comparison of the simula-tion and Poisson packet arrival process of latency and packet loss was acceptable and the jitter diverged when the load approached 1 (number of voice calls equals 1550). In both cases, as the load gets large (greater than 1), all performance measures converge.The difference in the jitter as determined from the simulation and Poisson was greater in the case of the T1 (40 ms range compared to 10 ms when using Pois-son), as compared to the T3 line (2.5 ms compared to 0.5 ms). So as the speed of the line increases, there seems to be a convergence in the absolute differences of the simulation and Poisson results. Thus, using a Poisson packet arrival model would be better on a backbone link than it would be on a low speed access link. For either speed, in the overloaded traffic cases—our main concern in this paper—the Poisson arrivals assumption appears to be appropriately based on these comparisons of the M/D/1/K queue versus the simulation of the Sriram and Whitt voice packet arri-val process. [8]Figure 7. Blocking Probability as Load Approaches 1Figure 8. Buffer Latency Comparison: Simulation and Poisson for T1Figure 9. Blocking Probability Comparison: Simulation and Poisson for T1Figure 10. Jitter Comparison: Simulation and Poisson for T1Figure 11. Buffer Latency Comparison: Simulation and Poisson for T3Figure 12. Blocking Probability Comparison: Simulation and Poisson for T3We have shown that the M/D/1/K analytics for jit-ter using equation (5) differed from the simulation of the Sriram and Whitt (1986) [8] voice packet arrival process. Our comments about equation (5) said that it was an approximation; therefore, an open question—“Was the analytic jitter approximation a significant contributor to the difference shown in Figures 10 and 13?” The answer to that question is no as is shown in Figure 14 and Table 4.ConclusionsIn this paper, we have looked at modeling the per-formance of voice and data packets in VoIP systems when the load gets large. We have presented analytic results for the voice and data packet latency, jitter, and loss probabilities for two VoIP queue disciplines. We have also studied the appropriateness of the Poisson voice packet arrival distribution. In addition, we have investigated using simulation models to study conges-tion in these systems. Modeling of a VoIP system using the WFQ discipline is an open area for future research.In conclusion, we have found that in developing simulation models for this type problem, the following is true.Runtimes can be extensive.Warm-up periods should be used to calculate jit-ter.GPSS/H or MS VBA models can have excessive runtimes. For a very simple single server system with a large number of packets (such as the OC-48 case), there may be some runtime efficienciesin using a general purpose (non-simulation) pro-gramming language. When fewer packets are inthe system (e.g., the T3 case), a simulation pack-age may be more efficient.The analytic results for the voice packet perform-ance are good when compared with simulation. They require significantly less runtime and agree with the simulation results. The latency and loss probability results are very good. The analytic jitter results are consistent with the simulated jitter in most cases. The analytic results for data packets are not as good as they were for voice packets, but are more than acceptable. These performance algorithms for individual links can be implemented for an analysis of a network, to esti-mate what a user’s experience will be with converged VoIP systems.AcknowledgmentsThis work was partially funded by the National Sci-ence Foundation (NSF) Grant Number DMII-0140232, the National Communications System (NCS) Contract Number NBCH-D-02-0039 (Task Order Number D020*******) with Mitretek Systems, and by Mitretek’s internally funded research program. References1.Joshi, M., A. Mansata, S. 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