Traffic modeling and analysis of hybrid fiber–coax systems1

Computer Networks and ISDN Systems 30 Ž1998. 821–834

Traffic modeling and analysis of hybrid fiber–coax systems David J. Houck 2 , Wai Sum Lai

1

)

AT & T Labs, 101 Crawfords Corner Road, Holmdel, NJ 07733-3030, USA

Abstract In the hybrid fiber–coax ŽHFC. architecture, the coax is a shared medium to which the network interface unit ŽNIU. of an end-user is attached for accessing a diversity of network services such as voicegrade and video telephony. For each call request from an end-user, connectivity to the network switch is established through a unified infrastructure. The traffic carrying capacities of the coax must therefore be understood so that the system can be dimensioned properly to ensure that a satisfactory level of blocking performance can be attained. This paper provides a high-level overview of an object-oriented simulation tool developed to model the HFC architecture. Key issues related to traffic capacities of the coax are analyzed both by using this tool and by analytical methods. q 1998 Elsevier Science B.V. Keywords: Hybrid fiber–coax systems; Broadband access; Finite-source model; Blocking probability; Call retries; Traffic simulation; Performance evaluation

1. Introduction The hybrid fiber–coax ŽHFC. architecture is a wire-based broadband access system that allows the network to deliver voicegrade, broadcast video, video dial-tone, and other services through a unified infrastructure. While fiber optics has the capacity needed for a broadband system, copper remains a cost-effective means for residential service distribution systems. HFC is a hybrid design that merges the best characteristics of both broadband and narrowband networks. Currently, the Asynchronous Transfer

) Corresponding author. Tel.: q1-732-9490268; fax: q1-7329491720; e-mail: [email protected]. 1 Part of this paper was presented at the Hybrid Fiber – Coax Systems Conference (Photonics East ’95), SPIE Proceedings, vol. 2609, Philadelphia, Pass., 23–24 October 1995, pp. 243–250. 2 Tel.: q1-732-9491290; fax: q1-732-9491720; e-mail: [email protected].

Mode ŽATM. Forum’s Residential Broadband ŽRBB. Working Group, the IEEE 802.14 Committee, and other standards bodies are in the process of developing interoperability standards Že.g., medium access control protocol w1,2x standard. that are applicable to HFC and related access systems such as Fiber To The Curb ŽFTTC.. In the HFC architecture, the coaxial cable Žcoax. is a shared medium to which the network interface unit ŽNIU. of an end-user is attached for accessing network services. For each call request from an end-user, connectivity to the network switch must be established. That is, the host digital terminal ŽHDT. located in the central office must allocate time slots on a demand basis for each connection on the downstream and upstream radio frequencies carried by the coax. The traffic carrying capacities of the coax must therefore be understood so that the system can be dimensioned properly to ensure that a satisfactory level of blocking performance can be attained. In this paper, work done on the modeling and

0169-7552r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 1 6 9 - 7 5 5 2 Ž 9 7 . 0 0 1 2 6 - 8

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D.J. Houck, W.S. Lai r Computer Networks and ISDN Systems 30 (1998) 821–834

analysis of the coax traffic capacity is presented. It first provides a high-level overview of an object-oriented event-driven simulation tool developed to model the HFC-2000e 3 Broadband Access System of Lucent Technologies ŽBell Laboratories Innovations.. One objective of this tool is to help estimate the traffic capacity of fiber nodes for telephony applications and to explore various alternatives in architecture and design. Another objective is to provide guidelines for the use of different time slot assignment algorithms and for capacity planning. To achieve these goals, salient operational and serving features of the HFC-2000e have been incorporated into the tool. As a preamble to the discussion of these features, an architectural overview of general HFC systems is presented in Section 2. For more details, the reader is referred to the system architecture overview presented in w3x. Section 3 describes the various features included in the simulator, together with a description of the simulator software structure. Section 4 deals with traffic modeling and assumptions while the user interface to the simulator is presented in Section 5. Section 6 illustrates the capabilities of the tool through examples which show the impact of architectural and algorithmic decisions. Analytical modeling is covered next, with Section 7 on burst proximity and Section 8 on the impacts of call retries.

2. Architectural overview We concentrate on the traffic carrying aspects of the system architecture for network access via HFC as pictured in Fig. 1. Žw4x contains a collection of articles on various aspects of HFC systems.. The path on the fiber from the fiber node to the host digital terminal ŽHDT. is dedicated capacity with no concentration or blocking. The path from the HDT to the local digital switch ŽLDS. is a Bellcore-specified TR-303 trunk group w5x. It is a full-access trunk group in which every call has access to every trunk and is a well-understood configuration. Therefore, the portion of the access network from the fiber node up was not included in the traffic study reported in

Fig. 1. System architecture.

this paper. The simulation was designed to capture the relevant traffic characteristics of the system by only modeling from the fiber node on down. There are two directions to consider: downstream Žfrom the LDS to an NIU. and upstream Žfrom an NIU to the LDS.. 4 They operate differently from each other. In the downstream direction, which operates in a broadcast mode, calls must obtain a time-division multiplexed ŽTDM. time slot on a radio frequency ŽRF. carried by the coax. In this direction, the NIU receiver is semi-permanently assigned to a particular frequency and thus at call setup time only time slots within that frequency are available. The NIUs that share a downstream frequency, but not necessarily a coax, are referred to as an NIU group. Since these NIU groups are semi-permanently defined, load balancing becomes a performance issue for the simulation to evaluate. In the upstream direction, calls must obtain a time

4

3

Trademark of Lucent Technologies.

The downstream and upstream channels are also referred to as the forward and reÕerse channels, respectively.

D.J. Houck, W.S. Lai r Computer Networks and ISDN Systems 30 (1998) 821–834

slot on an RF on the coax using a time-division multiple access ŽTDMA. protocol. The transmitter in an NIU may use several different RF frequencies, depending on the equipage at the HDT. This capacity is shared by all the NIUs on the same coax and allows for significant concentration which of course means a potential for blocking. Transmitters in the NIUs are agile enough to dynamically change frequencies and handle simultaneous calls on different frequencies in non-overlapping time slots. One of the goals of the simulator was to assess the impact of different spacings Žto be defined as burst proximity in Section 3. required between time slots on different frequencies. As to be described in the next section, time slots in the upstream direction do not form a full-access system. The simulation was used to assess the impact of this on the blocking performance. A few NIU types will be available, ranging from the single-living unit NIU with one or two lines up to a high-traffic multi-living unit NIU with many lines. It was not clear at the outset whether the higher traffic NIUs will need a second transmitter to achieve acceptable blocking. The simulation was useful in understanding the interaction between lines on the same NIU as well as between NIUs that share the same RF resource in terms of blocking probabilities.

3. Simulator overview 3.1. Operational and serÕice features The simulator models the traffic and serving characteristics of an entire fiber node with multiple coaxes, each carrying traffic both upstream and downstream. The following capabilities were incorporated in the simulator to evaluate the effects of various design alternatives: Ž1. Each NIU configuration can be varied in terms of the number of nailed-up and dynamically-assigned ports per NIU and the number of transmitters and receivers equipped on each NIU. An NIU transmitter is capable of hopping among up to a fixed number of frequencies, while an NIU receiver is tuned to one fixed frequency. Ž2. Any fixed number of RF channels per coax in the upstream direction, with a fixed number of DS0

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time slots 5 per channel, and restriction in frequency hopping for each NIU transmitter. The restriction applies to both nailed-up and dynamically-assigned ports. Note: As it takes time to hop frequencies, there is a restriction to the simultaneous access of different frequencies by a given single-transmitter NIU for time slot assignment. The number of DS0 time slots it takes an NIU transmitter to change RF channels is referred to as the burst proximity. For example, a burst proximity of 1 would mean that a transmitter using, say, time slot i in one frequency would be unable to use time slots i y 1, i, or i q 1 in any other frequency. A burst proximity of 0 means that an NIU can be allocated any non-overlapping time slot position in all the simultaneously-accessible frequencies. As shown later, burst proximity has an impact on the blocking performance. Ž3. Any number of RF channels shared among all coaxes in the downstream direction, with a fixed number of DS0 time slots per channel. There is no frequency hopping by the NIU receiver, i.e., an NIU receiver can read the time slots carried in one RF channel only. Ž4. Bi-directional DS0 and H0 service requests: A DS0 service Že.g., POTS, data. requires a single DS0 time slot upstream and a single DS0 time slot downstream. An H0 service, e.g., video telephony, requires an H0 upstream burst Žthe equivalent in duration of four contiguous upstream DS0 time slots. and six not necessarily contiguous downstream DS0 time slots. ŽAlthough downstream slots are not necessarily contiguous, they do have to maintain the correct sequence.. ISDN BRA service is provided by a nailed-up D-channel Žup to four multiplexed in a DS0. and dynamically-assigned B-channels ŽDS0 each.. ŽNote: In the rest of this paper, the number of downstream time slots required by a service is referred to as the batch size of the service.. Ž5. Since an NIU could potentially obtain an upstream H0 channel Ž4 contiguous time slots. and place up to 6 DS0 calls on it, this could be a useful feature for high-traffic NIUs. In this paper we refer to this as the phase arriÕal mode of operation. For

5 These are actually upstream bursts that carry 64-kbrs signals. For convenience, they are loosely referred to as DS0 time slots in this paper.

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D.J. Houck, W.S. Lai r Computer Networks and ISDN Systems 30 (1998) 821–834

NIUs operating in this mode, the first Žmodulo 6. DS0 service request will be assigned an H0 channel in the upstream direction and a single DS0 slot in the downstream direction. Subsequent Žfor up to a total of six simultaneous. calls from the same NIU would be placed in this H0 upstream burst. How to guarantee an efficient use of such H0 bursts was one of the issues that had to be resolved. Although not further discussed here, an engineering methodology for H0 bursts is presented in w6x. Ž6. Since different service requests may interfere with each other, the algorithms for assigning the RF resources in the two directions of transmission have a significant impact on the resulting blocking probability. The simulation models many such assignment algorithms for the different service types. To facilitate the evaluation of performance under different traffic scenarios, the simulator allows traffic characteristics to be specified on a per-port basis in terms of a large variety of service types. More will be said about this in Section 4. 3.2. Software description The tool is based on discrete-event simulation technique and is built using the object-oriented fea-

ture of Cqq. Objects are used to represent a broad spectrum of system components. These include physical components such as an NIU or a port, as well as conceptual elements such as a random number generator. Associated with each object is a set of properties called attributes. By allowing user-input data to set or modify the values for some of these attributes, the behavior or function of the objects can be specialized to accommodate the requirements of the user-specified network configuration that is to be simulated. Fig. 2 shows the architectural structure of the code. In this diagram, ovals are used to represent the different objects. A solid line between two ovals indicates that the two objects are in the same software module. The meanings of the dotted lines and the boxes are described following object description. There are five major objects: Meta-system, Call Arrivals, Service Manager, Event Processor, and Random Stream. In more details: Meta-system. Initializes the attributes contained in the data structures of various objects by taking data from a user-specified input data file. Interacts with the Event Processor object during the simulation interval while the latter is processing the different eÕents generated by the Call Arrivals and Service

Fig. 2. Simulation software structure.

D.J. Houck, W.S. Lai r Computer Networks and ISDN Systems 30 (1998) 821–834

Manager objects. Based on the Calendar Clock Žwhich is synchronized with the Clock object of the Event Processor., initiates the reporting of statistics at user-specified intervals. Call ArriÕals. Coordinates the activities of the following objects: Ø NIU, which keeps data about the number of nailed-up and dynamically-assigned ports in the physical NIU that it represents. Ø Port, which maintains the state Židle or busy. of the physical port it represents. As the NIU and the Port are tightly coupled, they are enclosed by a box as shown in Fig. 2. Ø SerÕice Type, which records the service attributes Ži.e., mean service time and standard deviation. of the service type it represents. All of these objects also collect statistics on the call durations, and the number of call arrivals and blocking. SerÕice Manager. Performs the allocation and deallocation of the SerÕer objects upon call arrivals and departures. It is also responsible for collecting statistics on the occupancy and number of calls served by the Server objects. Each SerÕer object represents a DS0 burst. These objects are grouped in a manner corresponding to the grouping of DS0 bursts within an RF frequency. EÕent Processor. Controls the following objects: Ø Event Ø Event List Ø Clock An ArriÕal EÕent object is generated by the Call Arrivals object each time a call arrives. Similarly, a Departure EÕent object is generated by the Service Manager object each time a call completion occurs. As these Event objects are generated, the Event Processor object puts them in the Event List object according to their chronological order. When the Event Processor object finishes processing an event, it removes the first Event object from the head of the Event List object and advances the time of the Clock object to correspond to the time of the Event object that is removed from the list. It can be seen from this description that, for each call, the Event object serves as a logical link between the Call Arrivals object and the Service Manager object via the Event Processor object. Such coupling

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is illustrated by the dotted lines that connect these objects in Fig. 2. Random Stream. Generates the stream of random numbers required to support the simulation. Different instantiations of the Random Stream object are used to provide the different independent arrival streams and service completion streams. This is to facilitate the reproducibility of simulation experiments. Inheritance plays a central role in the design of the simulator. It has been used extensively to specialize the behavior of different objects to reflect their real-world characters. For example, upstream and downstream time slots are managed differently as described previously in Section 2. Yet, they share a number of common operations, e.g., maintenance of idlerbusy states, association with assigned ports, use of time-slot assignment algorithm, collection of statistics, etc. Therefore, a base class of Service Manager objects is defined with Upstream Service Manager and Downstream Service Manager classes derived therefrom. As another example, a base Random Stream class has been implemented such that each instantiation uses the well-known linear congruential algorithm w7x to generate non-negative floating-point pseudorandom numbers uniformly distributed over the interval Ž0.0, 1.0x. Based on this, a set of derived classes has been created so that random variates with different distributions can be generated.

4. Traffic modeling A major goal in the design of the simulator is to reflect, as accurately as possible, the real traffic characteristics in the actual operating environment. In HFC systems, each port can access a diversity of service types and has its own service mix. So, each port must be characterized individually. To facilitate this, the simulator must allow traffic data to be specified on a per-port basis in terms of a large variety of service types. In a nutshell, this is done as follows. It is assumed that the mean service time Ži.e., call holding time. of each service type is known. Associated with each port is a serÕice profile, which specifies the probabilities for the port to generate the different service

D.J. Houck, W.S. Lai r Computer Networks and ISDN Systems 30 (1998) 821–834

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types. From this, the overall mean service time of the service requests from each port can be computed. Given the traffic loading of a port, the mean arrival rate of service requests from the port can then be determined. More details now follow. 4.1. Call arriÕals A finite-source quasi-random input process is used to characterize the call arrivals from the different ports, each modeled as an independent traffic source. In such a process, the probability of an arrival at a random time epoch depends directly on the number of idle Žand therefore, available to generate new calls. sources in that epoch. For any source that is idle in a given epoch, the time until the source next generates a call is exponentially distributed. When the number of lines on a coax is small, which is normally the case, this model captures the arrival patterns to the coaxes in an HFC system more accurately than a Poisson process. The latter is essentially an infinite-source input process. 4.2. Offered load For a finite-source model, input load can be specified in several ways. These are discussed below using the following notation: mean call holding time 1rm lˆ mean request rate of a source per unit of the source’s idle time l mean request rate of a source per unit of total time aˆ s lˆ rm offered load per idle source a s lrm offered load per source aX intended offered load per source m number of sources n number of servers The request rates lˆ and l are related. Assuming that blocked calls are cleared, the relation can be readily obtained by considering the mean interarrival time of call requests from a source: 1

l

1 s

lˆ

q Ž 1 y Pn . P

1

m

From this, the following pair of equations can be derived: aˆ a as , aˆ s . 1 q aˆ Ž 1 y P n . 1 y aŽ 1 y Pn . Note that both the offered load per source, a, and the offered load per idle source, a, ˆ are dependent on P n . Another parameter for load specification, the intended offered load, is congestion-independent in that it assumes there is no interference among the sources. That is, there are ample Ž n s m. servers so that all calls are served and P m s 0. In this specific case, let a X sa

P m s0 .

Then, X

aˆ

aX

, aˆ s . 1 q aˆ 1yaX Communications systems are usually engineered to have low blocking, say, in the neighborhood of 1%. The intended offered load can therefore be used as an approximation for the offered load. In the simulator, the traffic loading offered by a given port is taken to be the intended offered load of the port. This is specified by the hourly CCS Žhundred call seconds. 6 for that port. One then uses this per-port CCS and the per-port service profile to determine the mean arrival rates of the different service types at the port as follows. Let there be a total of k service types with service rates m i , i s 1, 2, . . . , k, respectively. If the service profile of a port is specified by the probabilities pi ’s, then the mean call holding time for the port is k p 1 i sÝ . m is1 m i a s

The intended mean arrival rate for the port is lX s a Xm , and the intended mean arrival rate of service type i is pi lX . 4.3. Simulation of finite-source model To simulate efficiently the finite-source effect of call arrivals, a thinning process is used in the simu-

,

where P n is call congestion, or the long-run proportion of lost calls.

6 The ‘hundred call seconds per hour’ ŽCCS. is a commonly used unit of traffic in telephone operating companies. There are 36 CCS in 1 Erlang.

D.J. Houck, W.S. Lai r Computer Networks and ISDN Systems 30 (1998) 821–834

lator. The maximum call arrival rate for all the sources is first determined by aggregating the call arrival rates of the individual sources, assuming all are idle. At each call arrival epoch, the next arrival epoch is generated according to the exponential distribution at this maximum rate. Each arrival event is associated with a source probabilistically, according to the individual rate of the source. An arrival event is accepted only if its associated source is idle at its arrival epoch; otherwise, the arrival event is rejected by simply throwing it away. Thinning the arrival stream in this way allows a single random number generator to be used for the combined call arrival process of all the sources, rather than having an independent random number generator for each source. This common random number generator achieves some level of variance reduction. Furthermore, as this scheme requires only one arrival event to be maintained in the event list, the average length of the event list is therefore relatively shorter. 4.4. Simulation of call holding time Recent studies by Bellcore w8,9x have found that call holding time distributions arise from a mixture of service Žcall. types each with a lognormal distribution. Thus, in the simulator, we model each service type by a lognormal distribution with its own mean and standard deviation. In conjunction with the use of a service profile, the desired mixture for call holding time distribution is thereby attained. 4.5. Analytic full-access model Because of burst proximity restrictions, time slots in the upstream direction do not form a full-access system. To benchmark the results obtained from the simulation tool, the Engset w10x full-access model will be used. One of our goals is to determine under what situations this or similar analytic models are sufficient for engineering purposes. Consider a full-access system with m identical sources and n servers, and an offered load per idle source of a. ˆ The Engset formula for time congestion, Pn , which gives the proportion of time during which all servers are busy during the busy hour, is Pn s

ž mn / aˆ

n n

Ý ks0

ž mn / aˆ

k

.

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The proportion of lost calls in the long run, as given by the Engset formula for call congestion, P n , is

Pn s

ž

m y 1 aˆ n n

/

n

Ý ks0

ž m yn 1 / aˆ

k

.

5. Use of simulator 5.1. Input specification Basic input data required include those that specify how long the simulation is to be run and how the output statistics are to be reported. In addition, the traffic model and the system configuration need to be specified. 5.1.1. Traffic data From the description in Section 4, input traffic data for the simulation tool are specified on three levels: Ž1. A set of service demands in terms of their respective batch sizes, means and standard deviations for the call holding times Že.g., facsimile calls, conversational calls and data calls represent different types of service demands with their own statistics.. Ž2. A set of service profiles that specifies the probabilities, i.e., the relative arrival rates, for the different service demands to be generated. Ž3. For each dynamically-assigned port in each NIU, the service profile used and the offered hourly CCS load. 5.1.2. System configuration The following input data describe the system configuration to be simulated: Ž1. The number of coaxes in the fiber node. Ž2. The number of RF channels and the number of DS0 time slots per RF channel in the downstream direction. Ž3. For each coax, the number of RF channels, the number of DS0 time slots per RF channel, and the burst proximity in the upstream direction. Ž4. For each coax, the number of NIUs, and for each NIU, the numbers of transmitters, receivers, nailed-up and dynamically-assigned ports. 5.2. Output statistics Currently, the following data are generated as output:

D.J. Houck, W.S. Lai r Computer Networks and ISDN Systems 30 (1998) 821–834

828 Table 1 Service type parameters Service type

Application

Batch size

Mean holding time Žs.

Standard deviation Žs.

1 2 3 4 5 6

facsimile conversational conversational telecommuting telecommuting video telephony

1 1 1 1 1 6

160 255 342 1800 3600 600

192 547 684 3600 7200 900

Ø The blocking probabilities for each NIU and each service type, separately in the upstream and downstream directions, and also combined for both directions. Ø The numbers of call arrivals, calls blocked Žeither upstream or downstream or both combined., the mean arrival rate and the mean call holding time for each NIU and each service type. Ø The occupancy of and number of calls served by each DS0 time slot, and also each RF channel, both upstream and downstream.

6. Simulation analysis and results

6.1.2. SerÕice profiles Four different service profiles are being modeled Žsee Table 2.. The first one is for a ‘normal’ call mix and associated holding time distribution; it has 10% type 1 Žfacsimile., 50% type 2 Žconversational., and 40% type 3 Žconversational. service. The second and the third represent a telecommuting line with a 50% chance of carrying a long holding time call. Since our simulator can give a different service profile to each port on an NIU, these profiles could be used to model the second line into a home that is used for telecommuting. The fourth profile is intended for a line that carries only video–telephony calls.

The simulation tool has been used with a set of hypothetical traffic data to analyze a wide variety of design alternatives and traffic engineering methodologies. In this section we illustrate the use of the simulator to show how burst proximity restrictions and video telephony impact blocking performance. As mentioned previously, there are many other factors that affect blocking performance. We will report on these in a future paper.

6.1.3. System configuration We model a hypothetical fiber node with 4 identical coaxes. We kept the number of RF channels in the upstream direction, the number of time slots per upstream channel, the number of RF channels in the downstream directions, and the number of time slots per downstream channel fixed over the set of simulation runs. The number, type, and traffic characteristics of the NIUs vary according to the factor studied and they are listed in the appropriate sections.

6.1. Hypothetical traffic model

6.2. Simulation results

6.1.1. SerÕice demands Six different service Žcall. types are being modeled: facsimile calls, two types of conversations, two types of long holding time telecommuting calls, and video–telephony calls. The characteristics of the first three service types are taken from the Bellcore data w8,9x. Those of the fourth and fifth are created to have, respectively, a 30- and a 60-minute mean holding time with a large variance. These are all DS0 services Žbatch size s 1. as shown in Table 1. The last service type is an H0 service Žbatch size s 6..

All results to be reported below are obtained after a 610-hour simulation run in which statistics were Table 2 Relative arrival rates of service types for each service profile Profile

1 2 3 4

Service type 1

2

3

4

5

6

0.1 0.1 0.1 0.0

0.5 0.2 0.2 0.0

0.4 0.2 0.2 0.0

0.0 0.5 0.0 0.0

0.0 0.0 0.5 0.0

0.0 0.0 0.0 1.0

D.J. Houck, W.S. Lai r Computer Networks and ISDN Systems 30 (1998) 821–834 Table 3 Variation of blocking with proximity rule

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collected starting at the 110th hour Žto preclude the initial transient period.. The values tabulated in what follows are per coax.

is further discussed in Section 7.. The fact that some entries in the table for proximity rule 0 show lower blocking than the one suggested by the Engset formula is ascribed to the statistical variation in simulation results. The above analysis of proximity effects is for single-slot applications such as voice telephony. Proximity restriction for multi-slot applications in the context of phase arrival mode of operation Žas defined in Section 3.1. is discussed in w6,11x. Proximity considerations for video telephony need further study.

6.2.1. Proximity rule sensitiÕity Because of burst proximity, any given multi-line NIU will not have full access to all the available slots in the different frequencies. This has impact on the blocking performance, as demonstrated below. For this set of simulation runs, we varied the proximity rule to determine the impact on blocking at various offered loads. We chose NIUs with 8 lines each and used different loads that correspond to different full-access ŽEngset. levels of blocking. These theoretical blocking levels are bounds on the best possible results for long-run averages with the proximity rule enforced. Table 3 shows the upstream blocking for proximity rules 0, 1, 2, and 3 and loads resulting in full-access blocking levels of approximately 0.5%, 1%, 5%, and 10%. The downstream blocking was negligible in all the examined cases. These results highlight two interesting points: Ž1. Proximity rules 0 and 1 yield approximately the same blocking and the increase is gradual as the proximity restriction increases to 3. Since with proximity rule 3 each active call is blocking the NIU from using 7 time slots in each of the other frequencies, one might have expected a larger impact. Ž2. The full-access Engset formula is a fairly accurate approximation of the actual blocking. ŽThis

6.2.2. Video telephony In the presence of different traffic types with different service requirements, it is important to assign time slots in a manner that minimizes the interference between different traffic types. In this example, we model a mixture of ordinary telephony traffic requiring DS0 time slots and 384-kbps video telephony ŽVT. requiring an H0 channel. High-traffic NIUs using the phase arrival mode exhibit some of the same characteristics. Here we chose a fixed number Ž15. of 8-line NIUs and a variable number Žfrom 20 to 60. of VT NIUs each of which is capable of carrying one VT call at a time. The average load is 6.9 CCSrline in an 8-line NIU and 3.0 CCSrline in a VT NIU. The ‘VT NIUs’ column in Table 4 shows the number of VT NIUs that coexist with the fifteen 8-line NIUs in a coax. The next three columns show the fractions of the VT load, the VT call arrivals during the simulation period, and the number of VT lines as percentages of the total load, the total call arrivals, and the total number of lines, respectively. The last three columns show the blocking experienced by the DS0 calls, the blocking experienced by the VT calls, and the overall average blocking experienced by all calls, respectively.

Average load Burst proximity ŽCCSrline. Engset 0

1

2

3

6.9 7.2 8.2 8.9

0.00486 0.01022 0.05286 0.10716

0.00595 0.01190 0.05593 0.11073

0.00792 0.01455 0.06147 0.11684

0.00475 0.00968 0.05024 0.10001

0.00440 0.00938 0.05098 0.10421

Table 4 Blocking for two traffic types VT NIUs

% Load

% Arrivals

% Lines

DSO blocking

Video blocking

Overall blocking

20 30 40 50 60

6.8 9.8 12.7 15.3 17.9

4.3 6.4 8.4 10.2 12.0

14.3 20.0 25.0 29.4 33.3

0.00001 0.00013 0.00065 0.00179 0.00257

0.00150 0.00398 0.01801 0.04579 0.08569

0.00008 0.00037 0.00210 0.00629 0.01256

830

D.J. Houck, W.S. Lai r Computer Networks and ISDN Systems 30 (1998) 821–834

These simulation results clearly reflect the standard multi-rate traffic engineering result that the wideband calls see significantly greater blocking. If this difference is unacceptable, then some form of capacity reservation scheme would be required.

7. Analytical modeling of burst proximity Burst proximity has the effect of limiting the availability of time slots in different upstream paths to any given single-transmitter NIU with multiple lines. In this section, the impact of this effect on the blocking performance is analyzed. 7.1. Scope and assumptions of analysis It must be stated at the outset that an exact solution of the proximity restriction is very difficult because of the apparent need to maintain an enormous state space to keep track of various dynamic interactions. The approach adopted here is similar to the average availability method of Bininda w12,13x used in the analysis of grading systems. The analysis is based on a modification of the full-access model presented previously. Simulation results are used as a guide to capture the impact of proximity restriction on the blocking performance. Consider a tagged NIU in the coax. One way to view its inaccessibility to certain time slots due to burst proximity, within the framework of the fullaccess model, is to pretend that these inaccessible DS0 packets are being occupied by traffic from other NIUs in the same coax. Along this line, the call congestion of the limited-access server system as seen by the tagged NIU can be approximated by the conditional call congestion of the same server system but with full access, given that the inaccessible time slots are occupied by other NIUs. Obviously, this viewpoint does not take the dynamic nature of time-slot assignments into account. As calls from different NIUs Žincluding the tagged NIU. come and go, different DS0 time slots change busyridle state at different times. So, the set of time slots inaccessible to the tagged NIU actually evolves over time. For the simple analysis intended here for engineering purpose, the effect of this dynamicity is being ignored.

7.2. Model for analysis Since the impact of proximity is on a per-NIU basis, a relevant quantity to be determined for each coax is the aÕerage number of time slot positions inaccessible to an NIU, n p . This will provide a basis for approximately evaluating the probability for time slots inaccessible to the tagged NIU to be occupied by other NIUs. The average total number of time slots assigned to all the NIUs is approximately given by the input load r from them. In terms of the load parameters previously defined in Section 4.2, r s ma X . With H NIUs and a burst proximity of p, n p f Ž2 p q 1. rrH. ŽNote that n p tends to increase with the number of RF channels. However, for a system with a small number of RF channels, we found by simulation that the dynamic effect of system evolution makes n p relatively stable.. The probability for these inaccessible time slots to be simultaneously occupied can be approximated by the time congestion of a full-access server system that has only n p servers, but fed with the same input traffic as that of the original server system Žwith n servers.. Let this probability be denoted by Pn p. Also, let P p and P be the call congestion of the limited-access and full-access server system with n servers as seen by an NIU, respectively. Then, from the discussion above,

P Pp f

Pn p

.

Using this formula, the limited-access blocking performance, P p , for the traffic scenario described in Section 6.2.1 are displayed in Table 5. It can be seen that these results are in general agreement with those in Table 3 from simulation, albeit using a very crude analytical model. It is acknowledged that the Table 5 Variation of blocking with proximity rule: Analytical results Average load Burst proximity Ž p . ŽCCSrline. Engset 0 1

2

3

6.9 7.2 8.2 8.9

0.00553 0.01126 0.05821 0.11556

0.00596 0.01212 0.06262 0.12423

0.00475 0.00968 0.05024 0.10001

0.00488 0.00994 0.05156 0.10256

0.00517 0.01054 0.05459 0.10848

D.J. Houck, W.S. Lai r Computer Networks and ISDN Systems 30 (1998) 821–834

model could be further improved. However, for engineering and planning purposes, it is adequate. 8. Analytical modeling of call retries In circuit-switched telecommunications networks, call retries can cause an increase in the loading offered to the network, resulting in the need to add extra capacity. For example, with probability of retries typically at 0.7 to 0.8, the load of a network under congestion can be increased by 3 to 5 times of the normal load w14x. In HFC systems, the total number of lines supported by a coax is much smaller than that of a network switch. A model that takes into account the finite number of traffic sources on each coax is developed in this section to analyze the increase in loading due to retries. In the literature w15,16x, the modeling of multiserver systems with retries usually assumes a Poisson input stream with an infinite number of traffic sources. To reduce to a finite system of balance equations, the general method is to upper-bound the number of sources generating retries to some given constant. In the problem at hand, the number of sources generating both new and retrial traffic is finite. This leads to a finite system of equations which is then solved exactly. Note that, to maintain tractability, a full-access system is assumed here. As demonstrated in the previous section, simple adjustment for non-full-access can be made, if necessary. 8.1. Model for analysis The following notation is used: m n

number of sources number of servers

lˆ 1rm a

831

request rate per idle source mean call holding time offered load per idle source Žs lˆ rm . Ž Note: The symbol aˆ was used in Section 4. For simplicity, a is used here.. retrial interval Žrelative to the unit mean 1rŽ rm . call holding time. R probability of retries i number in service in primary station j number in orbit station Pi, j probability that the system is in state Ž i, j . Consider a finite-source loss system with retries. There are m sources and n servers. An idle source generates calls Žrequests for service. at rate lˆ . At the time of request, if there are one or more free servers, the request is served immediately, with mean holding time 1rm. Otherwise, it is blocked. This is called an oÕerflow and there are two possibilities: Ø With probability 1 y R, the source makes a new request at the ordinary rate of lˆ Žthe original request is therefore lost.. Ø With probability R, the source returns and retries Ži.e., re-issues the request. at a rate faster than lˆ . For convenience, the retrial interÕal, or the mean delay between blocking and reattempt, is defined relative to the unit mean call holding time as 1rŽ rm . - 1rlˆ . A source that retries is said to be in orbit. The original n-server system is referred to as the primary station. At any time, there are 0 F i F n sources receiving service at the primary station and 0 F j F m y n sources in orbit. The operation of the system is depicted in Fig. 3. For a Markovian retrial system, i.e., one in which the interarrival times Žfrom an idle source., service times, and retrial intervals are exponentially dis-

Fig. 3. A finite-source loss system with retries.

D.J. Houck, W.S. Lai r Computer Networks and ISDN Systems 30 (1998) 821–834

832

tributed random variables, the state transitions are governed by the following equations: Ž m y i y j . a q i q jr Pi , j

This equation has the general solution w17x m

G Ž x , y . s Ž 1 q ax . c Ž u . , where c is an arbitrary function of u, with

s Ž m y i y j q 1 . aPiy1, j q Ž i q 1 . Piq1, j us yy

q Ž j q 1 . rPiy1, jq1 for 0F i - n and 0 F j Fm y n, Ž m y n y j . Ra q n q j Ž 1 y R . r Pn , j

Fi Ž y . s

1 Ei i! E x i myn

s

or j ) m y n,

=

hs0

Ý

1 k h y h ryay1

ž/ ž

ž

aj y

jsk

myn

Ý

`

rj

Ý hs0



1qa

`

myn

Ý Ý

Pi , j y j x i s

is0 js0

Ý Fi Ž y . x i . is0

Using these generating functions, the infinite set of state transition equations can be transformed to yield the partial differential equation: E GŽ x , y. ax 2 q Ž 1 y a . x y 1 Ex E GŽ x , y. q Ž r y a . y q Ž ay y r . x Ey s yma Ž 1 y x . G Ž x , y . .

/

jyk

rya ryay1

/

ž/

n

js0

GŽ x , y. s

ky h

j u Ž iyjqk. k j where a j , j s 0, 1, . . . , m y n, are unknown coefficients in the expansion of c Ž u., and =

uj Ž n. s

Pi , j y j ,

.

xs 0

myn

Pi , j s 1.

This is a finite system of simultaneous equations. To solve it, the finiteness of i is first ignored by assuming that the state transitions are valid for all i G 0, thereby obtaining an infinite set of equations. The following generating functions can then be defined:

Ž rr Ž1qa ..

GŽ x , y.

Ý Ý

is0 js0

Fi Ž y . s

y1q Ž rr Ž1qa ..

k

ks0

myn

Ý Ý

Ž1yx .y

By expanding c Ž u. in Taylor series, we obtain

for isn and 0FjFm y n, Pi , j s 0, for i - 0, or j - 0, or i ) N, n

ryay1

= Ž 1 q ax .

s Ž m y n y j q 1 . a Ž Pny 1 , j q RPn , jy1 . q Ž j q 1 . r Ž Pny 1, jq1 q Ž 1 y R . Pn , jq1 .

Ž r y a. x y 1

ž

h r

0 /

y 1 j ny h a 1qa h for j s 0, 1, . . . , m y n These coefficients, a j ’s, can be determined by using the boundary state transition equation for i s n, and the normalization condition, =



Mq

qhy1

0

Ý Pi , j s 1. i, j

The probability of blocking can be evaluated by FmŽ1..

Table 6 Sensitivity to retrial interval r 1rŽ rm . Blocking prob.

0.05 20.0 0.00358245

0.0820559 12.1868 0.00364314

0.1 10.0 0.00366380

1.0 1.0 0.00397611

10.0 0.1 0.00450782

r 1rŽ rm . Blocking prob.

15 0.0666667 0.00452973

20 0.05 0.00451959

100 0.01 0.00416542

1000 0.001 0.00372957

100,000 0.00001 0.00364407

D.J. Houck, W.S. Lai r Computer Networks and ISDN Systems 30 (1998) 821–834

833

Table 7 Effect of retries under high blocking Number of servers

21 20 19 18 17 16 15

With retries

No retries

% Load increase due to retries

sources

block prob.

sources

block prob.

155 155 155 155 155 155 155

0.00450782 0.00875925 0.0161666 0.0283203 0.0470757 0.0742857 0.11142

159 159 160 162 163 165 168

0.00471644 0.00872493 0.0160885 0.0289637 0.0467183 0.0731639 0.110029

As an example, consider a 21-server system with each idle source offering a load of 0.0820559 Erlangs s 2.95401 CCS. For a 0.5% blocking objective, this system can support 159 sources with a blocking probability Žtime congestion. of 0.00471644 when there is no retry. When there is a 0.7 retry probability and a retry interval of 10% of the call holding time, the time congestion degrades to 0.00589602. For the same set of system parameters, except with 155 sources, the blocking probability is 0.00364314 Žwith no retries. and 0.00450782 Žwith retries.. Thus, the system should be engineered to support no more than 155 sources to meet the service criterion, as there is a potential load increase of Ž159–155.r155 s 0.0258 due to retries. Thus, because of the small total number of lines supported by a coax, the increase in loading as a result of call retries is very small. It would be interesting to examine how time congestion varies with the length of retrial interval. Using the same scenario as above for 155 sources, we obtain the results Žwith m normalized to 1.0rs. given in Table 6. When the rate of retrial is the same as the request rate Ž0.0820559rs., the time congestion Ž0.00364314. is identical to the case when there are no retries. As r decreases so that the relative retrial interval increases, the time congestion decreases. This is to be expected since less overall traffic is generated in this case. In the opposite direction, as r increases, the time congestion rises to a peak around r s 15 and then decreases asymptotically towards 0.00364314, the value obtained when there are no retries. An intuitive explanation is that when the relative retrial interval is infinitesimally small, a

2.58 2.58 3.23 4.52 5.16 6.45 8.39

retry will exit the orbit and turn back into a source with probability 1. We now turn to determine the level of blocking at which retries could have a significant impact. Using the same system parameters as before, we gradually increase the blocking by reducing the number of servers. As shown in Table 7, the impact due to retries becomes significant only when the blocking well exceeds 2% .. As access systems typically operate at a level of blocking that is no more than 1%, the effect of retries can be neglected. 9. Conclusions An object-oriented simulation tool has been developed to investigate the performance of Lucent Technologies’ HFC-2000e Broadband Access System with the objective of estimating traffic capacity and exploring architectural alternatives. Salient operational and service features have been incorporated into the tool. Analyses using this tool indicate that burst proximity restrictions can increase blocking, but not as severely as one might expect. Based on these simulations results, a simple analytical model has been developed that can be used for engineering purposes. Modeling of call retries shows that, unlike network switches, the increase in loading due to retries for HFC systems is negligible because of the small number of lines involved. Acknowledgements The authors wish to thank Reed Even of Lucent Technologies for his careful description of the im-

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portant features of this architecture and even more careful filtering of unimportant issues. Comments from the anonymous referees helped clarified the text and suggested the numerical analysis for retries under variable retrial interval and under high blocking.

References w1x J.E. Dail, M.A. Dajer, C.C. Li, P.D. Magill, C.A. Siller Jr., K. Sriram, N.A. Whitaker, Adaptive digital access protocol: A MAC protocol for multiservice broadband access networks, IEEE Comm. Magazine 34 Ž3. Ž1996. 104–112. w2x B.T. Doshi, S. Dravida, P.D. Magill, C.A. Siller Jr., K. Sriram, A broadband multiple access protocol for STM, ATM, and variable length data services on hybrid fiber–coax networks, Bell Labs. Techn. J. 1 Ž1. Ž1996. 36–65. w3x B.A. Kaplan, Hybrid fiber–coax architecture overview, Hybrid Fiber–Coax Systems Conference ŽPhotonics East ’95., Philadelphia, Pa., 23–24 October 1995, SPIE Proc., vol. 2609, pp. 34–38. w4x W.S. Lai, S.T. Jewell ŽEds.., Hybrid Fiber–Coax Systems, SPIE Proceedings, vol. 2609, Philadelphia, Pa., 23–24 October, 1995. w5x W. Arvidson, S. Jiang, An integrated switching system interface for the telephony services of a hybrid fiber–coax system, Hybrid Fiber–Coax Systems Conference ŽPhotonics East ’95., Philadelphia, Pa., 23–24 October 1995, SPIE Proc., vol. 2609, pp. 80–91. w6x D.J. Houck, W.S. Lai, Engineering the upstream traffic capacity of hybrid fiber–coax systems, Proc. 15th Int. Teletraffic Congress ŽITC15., Washington, DC, 23–27 June, 1997, pp. 845–856. w7x A.M. Law, W.D. Kelton, Simulation Modeling and Analysis, 2nd ed., McGraw-Hill, New York, 1991. w8x V.A. Bolotin, Modeling call holding time distributions for CCS Network design and performance analysis, IEEE J. Select. Areas Commun. 12 Ž3. Ž1994. 433–438. w9x V.A. Bolotin, Telephone circuit holding time distributions, Proc. 14th Int. Teletraffic Congress, 1994. w10x R.B. Cooper, Introduction to Queueing Theory, CEEPress Books, Washington, DC, 1990.

w11x D.J. Houck, W.S. Lai, Traffic analysis of hybrid fiber–coax systems, Broadband Access Systems Conf. ŽPhotonics East ’96., Boston, Mass., 19–22 November 1996, SPIE Proceedings, vol. 2917, pp. 350–357. w12x N. Bininda, A. Wendt, The effective availability of serving trunk groups succeeding link arrangements, Nachricht Technishe Zeitung, 1959, pp. 579–585. w13x R. Mina, Introduction to Teletraffic Engineering, Telephony, Chicago, 1974. w14x P.K. Reeser, Simple approximation for blocking seen by peaked traffic with delayed, correlated reattempts, Proc. 12th Int. Teletraffic Congress, Torino, 1988. w15x G. Falin, A survey of retrial queues, Queueing Systems 7 Ž1990. 127–168. w16x T. Yang, J.G.C. Templeton, A survey of retrial queues, Queueing Systems 2 Ž1987. 201–233. w17x P.R. Garabedian, Partial Differential Equations, Wiley, New York, 1964. David Houck received his B.A. in Mathematics in 1970 and his Ph.D. in Operations Research in 1974, both from the Johns Hopkins University. Dave was an Assistant Professor at the University of Maryland Baltimore County from 1974–1979 and has been at AT&T ŽBell. Labs since 1979. His current activities include the modeling and performance analysis of systems and services developed by AT&T. Prior work at AT&T includes switch planning, operator services traffic studies, network design, implementation of Karmarkar’s linear programming algorithm, and airline crew scheduling algorithms. Wai Sum Lai received his B.Sc. with First Class Honours in Electrical Engineering from the University of Hong Kong in 1970, M.S. in Information and Computer Sciences from the University of Hawaii in 1974, and Ph.D. in Systems and Computer Engineering from Carleton University, Ottawa, Canada in 1987. His international professional experience includes work at Trans-World Electronics Ltd. in Hong Kong, the ALOHA System at the University of Hawaii, and Bell-Northern Research Žnow renamed Nortel Technology. in Ottawa, Canada. During 1987 and 1988, he led the T1S1.1 Group on Packet Mode Services that developed the ANSI T1.606-1990 ISDN Frame Relay standard. Since 1988, he has been with AT&T ŽBell. Labs in Holmdel, New Jersey, working on the performance modeling of systems and services developed by AT&T.