Connection admission control based on bandwidth and buffer usage
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 WO2000038380A2 WO2000038380A2 PCT/SE1999/002419 SE9902419W WO2000038380A2 WO 2000038380 A2 WO2000038380 A2 WO 2000038380A2 SE 9902419 W SE9902419 W SE 9902419W WO 2000038380 A2 WO2000038380 A2 WO 2000038380A2
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 traffic
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 H—ELECTRICITY
 H04—ELECTRIC COMMUNICATION TECHNIQUE
 H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
 H04L12/00—Data switching networks
 H04L12/54—Storeandforward switching systems
 H04L12/56—Packet switching systems
 H04L12/5601—Transfer mode dependent, e.g. ATM

 H—ELECTRICITY
 H04—ELECTRIC COMMUNICATION TECHNIQUE
 H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
 H04L12/00—Data switching networks
 H04L12/54—Storeandforward switching systems
 H04L12/56—Packet switching systems
 H04L12/5601—Transfer mode dependent, e.g. ATM
 H04L12/5602—Bandwidth control in ATM Networks, e.g. leaky bucket

 H—ELECTRICITY
 H04—ELECTRIC COMMUNICATION TECHNIQUE
 H04Q—SELECTING
 H04Q11/00—Selecting arrangements for multiplex systems
 H04Q11/04—Selecting arrangements for multiplex systems for timedivision multiplexing
 H04Q11/0428—Integrated services digital network, i.e. systems for transmission of different types of digitised signals, e.g. speech, data, telecentral, television signals
 H04Q11/0478—Provisions for broadband connections

 H—ELECTRICITY
 H04—ELECTRIC COMMUNICATION TECHNIQUE
 H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
 H04L12/00—Data switching networks
 H04L12/54—Storeandforward switching systems
 H04L12/56—Packet switching systems
 H04L12/5601—Transfer mode dependent, e.g. ATM
 H04L2012/5629—Admission control

 H—ELECTRICITY
 H04—ELECTRIC COMMUNICATION TECHNIQUE
 H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
 H04L12/00—Data switching networks
 H04L12/54—Storeandforward switching systems
 H04L12/56—Packet switching systems
 H04L12/5601—Transfer mode dependent, e.g. ATM
 H04L2012/5638—Services, e.g. multimedia, GOS, QOS
 H04L2012/5646—Cell characteristics, e.g. loss, delay, jitter, sequence integrity
 H04L2012/5651—Priority, marking, classes

 H—ELECTRICITY
 H04—ELECTRIC COMMUNICATION TECHNIQUE
 H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
 H04L12/00—Data switching networks
 H04L12/54—Storeandforward switching systems
 H04L12/56—Packet switching systems
 H04L12/5601—Transfer mode dependent, e.g. ATM
 H04L2012/5678—Traffic aspects, e.g. arbitration, load balancing, smoothing, buffer management
 H04L2012/5679—Arbitration or scheduling

 H—ELECTRICITY
 H04—ELECTRIC COMMUNICATION TECHNIQUE
 H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
 H04L12/00—Data switching networks
 H04L12/54—Storeandforward switching systems
 H04L12/56—Packet switching systems
 H04L12/5601—Transfer mode dependent, e.g. ATM
 H04L2012/5678—Traffic aspects, e.g. arbitration, load balancing, smoothing, buffer management
 H04L2012/5681—Buffer or queue management

 H—ELECTRICITY
 H04—ELECTRIC COMMUNICATION TECHNIQUE
 H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
 H04L12/00—Data switching networks
 H04L12/54—Storeandforward switching systems
 H04L12/56—Packet switching systems
 H04L12/5601—Transfer mode dependent, e.g. ATM
 H04L2012/5678—Traffic aspects, e.g. arbitration, load balancing, smoothing, buffer management
 H04L2012/5684—Characteristics of traffic flows
Abstract
Description
CONNECTION ADMISSION CONTROL BASED ON BANDWIDTH AND BUFFER USAGE
BACKGROUND
1. FIELD OF THE INVENTION The present invention pertains to telecommunications, and particular to connection admission control aspects of telecommunications traffic management.
2. RELATED ART AND OTHER CONSIDERATIONS
In telecommunications, traffic management is the art of providing users with the service they need and have paid for. One aspect of traffic management, connection admission control (CAC), checks to ensure that resource consumption of new connections will not violate the quality of service (QOS) requirements of new and existing connections before admitting the new connections on the network. Relevant resources involved in connection admission control (CAC) are channel numbers, bandwidth, and buffer space.
Within connection admission control (CAC), effective/equivalent bandwidth methods are based on the asymptotic behavior of a tail of a queue length distribution. The algorithms of such methods calculate an effective bandwidth based on the traffic descriptor, QOS requirements, and buffer resources. The effective bandwidth methods are often restricted to a certain type of traffic sources. Some results apply to onoff Markov fluids (see Kesidis, Walrand, Chang, "Effective Bandwidths for Multicall Markov Fluids and Other ATM Sources", IEEE Trans. Networking, Vol. 1, No. 4, pp. 424  428, August 1993), others to leakybucket shaped traffic, see Lo Presti, Zhang, "Source Time Scale and Optimal Buffer/Bandwidth Tradeoff for Regulated Traffic in an ATM Node", UMASS CMPSCI Technical Report UMCS9638.
Some real sources may be closest to the Markov fluidtype model, for instance encoded speech. This type of source needs no further shaping to be suitable for transfer through a network, although it has no maximum burst size. On the other hand, packet traffic is typically leaky bucket shaped with a peak rate, a sustainable rate, and a maximum burst size. The asymptotic behavior of a queue loaded with leaky bucket traffic differs significantly from the behavior of a queue loaded with Markov traffic. The leaky bucketloaded queue has an upper bound on its length, while the Markov loaded queue has no upper bound. Using the asymptotic Markovtype results on leaky bucket traffic will completely loose the leaky bucket burstiness information. Using the leaky bucket results with a Markov source does not work because they rely on the limited burst size.
What is needed, therefore, and an object of the invention, is a connection admission control (CAC) technique which is effective in multitudinous traffic scenarios.
BRIEF SUMMARY OF THE INVENTION
The present invention approximates probability of loss using a log moment generating function and its two partial derivatives of workload on a queue over a time interval. The approximation uses four state variables, which depend on the log moment generating function and its two partial derivatives. The four state variables are as follows:
z(s,t) Linear term in approximation to log loss ratio at working point (s,t)
c(s,t) Argument of logarithmic term in approximation to log loss ratio at working point (s,t)
B(s,t) Buffer limit used at working point (s,t)
m(s,t) Multiplier of imaginary traffic used at working point (s,t)
Advantageously, these state variables vary linearly with the traffic, so a new connection can simply add its contributions to them.
Finding the point (s,t) which optimizes the approximation to the probability of loss with the actual buffer size limit, service, and traffic, in real time is generally not feasible. Instead, the present invention uses a predetermined working point (s,t) and adds some imaginary traffic from a design traffic mix to the actual traffic in order to make the working point the optimizing point for the sum of real and imaginary traffic. This makes it sufficient to keep track of the state variables rather than of whole functions.
The approximation of the invention uses the state variables to produce the following three parameters: (1) an approximation q = z  log(c) to the logarithm of the probability of loss; (2) a buffer size limit B; and (3) a multiple m of imaginary traffic from a design mix.
The traffic on all connections is admissable if the four conditions are satisfied. The first condition is that q be less than or equal to the QOS log loss requirement. The second condition is that B be less than or equal to the limit set by available buffer space and QOS delay requirements. The third condition is that m is nonnegative. The fourth condition is that the mean input rate of real plus imaginary traffic exceeds the mean service rate by no more than admitted by the QOS loss requirement.
If none of the first three conditions is satisfied, the traffic is inadmissible. If some, but not all, of the conditions are satisfied, the algorithm cannot make a determination, and therefore, does not admit the traffic.
To find a good working point, the above calculations are performed offline for a great number of points (s,t), and a working point is picked from a region in the (s,t) plane that performs best with the design traffic mix. If it is desired to widen the tolerance against deviations from the design mix, such may be possible by choosing a working point that is not optimal for the design traffic mix. If it is desired to handle several design traffic mixes, a good working point can be selected for each design mix, state variables maintained for each working point, and calculate q, B, and m calculated at each working point at connection setup. If no working point rejects the traffic, and at least one working point admits the traffic, the traffic is deemed admissible.
The present invention applies, e.g., to a single queue and server. Moreover, the invention can be generalized to multiple queues and servers.
The present invention has a wide range of applicability, working both on a mixture of encoded speech and leaky bucket shaped traffic as well as other types of traffic. The traffic may be of fluid or discrete arrivals type. In examples with known exact answer considered so far the method has admitted between 97% and 100% of what is theoretically admissible. Although more complex than a pure efficient bandwidth method, the invention is implementable in real time.
BRIEF DESCRIPTION OF THE DRAWINGS
The foregoing and other objects, features, and advantages of the invention will be apparent from the following more particular description of preferred embodiments as illustrated in the accompanying drawings in which reference characters refer to the same parts throughout the various views. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating the principles of the invention.
Fig. 1 is a graph illustrating causes of underestimating loss.
Fig. 2 is a graph illustrating bounding of x  B by ke^{sx}
Fig. 3 is a schematic view illustrating a highpriority queue H and a lowpriority queue L served by a single server.
Fig. 4 is a schematic view illustrating N queues of equal priority served by a single server.
Fig. 5 is a graph showing a simulation of queue length distributions.
Fig. 6 is a schematic view of an example of queues with shared buffer and independent servers.
Fig. 7 is a schematic view of a queueing system in a spatial switch of a wideband
CDMA telecommunications system.
Fig. 8 A is a graph of a log moment generating function; Fig. 8B is a graph of the derivative with respect to s of the log moment generating function of Fig. 8 A; Fig. 8C is a graph of the derivative with respect to t of the log moment generating function of Fig. 8 A. Fig. 9A  Fig. 9E are graphs showing ONOFF periodic arrivals.
Fig. 10A is a graph showing the log moment generating function, and Fig. 10B and Fig. IOC its derivatives with respect to s and t, respectively, for ONOFF periodic source with R=l, T=l, T_{on}=0.2.
Fig. 11 A is a graph showing the log moment generating function, and Fig. 1 IB and Fig. 1 IC are graphs showing the derivatives of the log moment generation function of Fig. 1 1A with respect to s and t, respectively, for discrete periodic source with T=l, R=l.
Fig. 12A is a graph showing the log moment generating function, and Fig. 12B and Fig. 12C are graphs showing the derivatives of the log moment generation function of Fig. 12A with respect to s and t, respectively, for a Poisson source with T=l, R=l.
Fig. 13 A is a graph showing the log moment generating function, and Fig. 13B and Fig. 13C are graphs showing the derivatives of the log moment generation function of Fig. 13A with respect to s and t, respectively, for an ON/OFF Markov fluid source with T=l/Q_{1>2}+1/Q_{2ι l}=l, T_{on}=l/Q_{1>2}=0.2, R=l.
Fig. 14A is a graph showing the log moment generating function, and Fig. 14B and Fig. 14C are graphs showing the derivatives of the log moment generation function of Fig. 14A with respect to s and t, respectively, for always ON or OFF fluid source with P_{0N} = 0.2 and R = 1.
Fig. 15A and Fig. 15B are graphs showing contours of q, B, and m for traffic cases 1, 10, respectively.
Fig. 16A  Fig. 16D are graphs showing contours of q, B, and m for traffic cases 2, 3, 4, 5, respectively.
Fig. 17A  Fig. 17D are graphs showing contours of q, B, and m for traffic cases 6, 7, 8, and 9, respectively. Fig. 18A  Fig. 18D are graphs showing contours of q, B, and m for traffic cases 11, 12, 13, and 14, respectively.
Fig. 19A  Fig. 19D are graphs showing contours of q, B, and m for traffic cases 15, 16, 17, and 18, respectively.
Fig. 20 is a graph showing contours of q, B, and m for a 19^{th} one connection traffic case.
Fig. 21A  Fig. 21D are graphs showing contours of q, B, and m for traffic cases 20, 29, 39, 48, respectively.
Fig. 22A  Fig. 22D are graphs showing contours of q, B, and m for traffic cases 21 , 22, 40, 41 , respectively.
Fig. 23 A  Fig. 23D are graphs showing contours of q, B, and m for traffic cases 23, 24, 42, 43, respectively.
Fig. 24A  Fig. 24D are graphs showing contours of q, B, and m for traffic cases 25, 26, 44, 45, respectively.
Fig. 25 A  Fig. 25D are graphs showing contours of q, B, and m for traffic cases
27, 28, 46, 47, respectively.
Fig. 26A  Fig. 26D are graphs showing contours of q, B, and m for traffic cases 30, 31, 49, 50, respectively.
Fig. 27A  Fig. 27D are graphs showing contours of q, B, and m for traffic cases 32, 33, 51, 52, respectively.
Fig. 28A  Fig. 28D are graphs showing contours of q, B, and m for traffic cases 34, 35, 53, 54, respectively.
Fig. 29A  Fig. 29D are graphs showing contours of q, B, and m for traffic cases 36, 37, 55, 56, respectively. Fig. 30A  Fig. 30B are graphs showing contours of q, B, and m for traffic cases 36, 38, 57, respectively.
Fig. 31 is a schematic view of an node according to an embodiment of the invention.
Fig. 32A is a schematic view of an node entity which includes a node main processor.
Fig. 32B is a schematic view of an node entity which serves as an extension terminal.
Fig. 33 is a graph showing a probability density function.
DETAILED DESCRIPTION OF THE DRAWINGS
1 Introduction
Link Admission Control LAC is part of Connection Admission Control CAC. LAC examines whether resources are available for a new connection on a link. Resources are channel numbers, buffer space and bandwidth. The present invention considers LAC with respect to bandwidth and buffer usage only.
A link may contain several queueandserver systems, for instance, egress queues in the sending node and ingress queues in the receiving node. LAC must check for resources in every such system. The present invention illustrates an exemplary system.
In the above regard, Fig. 31 shows a representative node 20 with which the connection admission control (CAC) technique of the present invention can be employed. The particular node 20 which serves as an illustration of the connection admission control (CAC) of the present invention is an Asynchronous Transfer Mode (ATM) node 20 comprising a switch core 24. Switch core 24 which has plural switch core ports, four of the switch core ports being shown as switch core ports 26 A  26D in Fig. 31. A node entity 30, also known as a device board, is connected to each of the switch core ports. Fig. 31 shows node entity 30A being connected by a bidirectional link 32 A to switch core port 26A; node entity 3 OB being connected by another bidirectional link 32B to switch core port 26B; and so forth. It should be understood that more than four node entities 30 can be, and typically are, connected to corresponding ports 26 of switch core 24, but that only four node entities 30 are shown for sake of simplification.
Each node entity 30 performs one or more functions and has, among other components hereinafter described, a processor mounted thereon. One of the node entities 30, particularly node entity 30A, has a node main processor which generally supervises operation of the entire ATM node 20. The other node entities 30, such as node entities 30B  30D, have entity processors 50B  50D, respectively, also known as board processors.
In the particular embodiment shown in Fig. 31, each of node entities 30B  30D serve as extension terminals. Having such function, the node entities 30B  30D are connected by physical lines or links to other ATM nodes. For example, node entity 30B is shown as having four physical lines 60B1 through 60B4 to other (unillustrated) ATM node(s). Although not necessarily labeled, in Fig. 31 , the other node entities 30B and 30C also have four physical lines extending to other (unillustrated) ATM node(s).
In general, the ATM node 20 serves to route ATM traffic cells between physical lines 60 which connect ATM node 20 to other ATM nodes. For example, ATM traffic cells incoming to ATM node 20 on physical line 60B1 can be routed by switch core 24 to be outgoing from ATM node 20 on physical line 60C1. The entity processor of each node entity 30 plays a significant role when establishing ATM connections to/from that entity. In case of an extension terminal (ET) entity, the establishing of an ATM connection between a physical line and another node entity (e.g., another extension terminal or any other type of node entity) is performed by setting up a translation table row (hosted in the ATM line module), one for each direction. In the ingress direction, the translation assigns an internal VP CI and an addressee switch port for each utilized VPI/VCI on the physical link. The addressee switch port is used to route each cell to the right switch port (i.e., node entity). In the egress direction, the translation assigns the VPI/VCI to be used on the physical link for each VPI/VCI used internally between two node entities. When actually transmitting cells on the connection, only hardware is involved (e.g., no processors perform any tasks concerning cell transfer). In the case of any other type of node entity (e.g., an entity that terminates an ATM connection), the principles discussed above apply except for the egress direction in which no external VPI/VCI is assigned. Instead, a termination point (software entity of the processor) is utilized.
As mentioned above, the overall operation of ATM node 20 is managed by node main processor 40. In order to communicate with the node entities 30, and particularly with the entity processors 50 of the respective node entities 30, certain control paths must be established between node main processor 40 and the entity processors 50 so that the processors can communicate with one another. The communication is performed by cells which are transmitted over the control paths established between node main processor 40 and the various entity processors 50. Establishment of these control paths is understood with reference to U.S. Patent Application SN 09/249,785 filed February 16, 1999, "Establishing Internal Control Paths in ATM Node," which is incorporated herein by reference.
Fig. 32A shows an example node entity 30A at which node main processor 40 is situated. The node entity 30 of Fig. 32A includes a switch port interface module (SPIM) 30A1 which is connected by bidirectional link 32A to switch core 24. The switch port interface module (SPIM) 30A1 is connected to bus 30A2, which is preferably a UTOPIA standard bus. The node main processor 40 is connected by bus 30A2 to switch port interface module (SPIM) 30A1. As discussed herein, and particularly illustrated in Fig. 7, the switch port interface modules (SPIMs) have queues which are affected by traffic management.
Fig. 32B shows an example node entity 30 which serves as an extension terminal. Like Fig. 32A, the node entity 30 of Fig. 32B has switch port interface module (SPIM) 30B1 and bus 30B2, with a processor (entity processor 50) being connected to bus 30B2. In addition, bus 30B2 is connected to ATM line module 30B 3. The ATM line module 30B3, as hereinafter explained, contains VPI/VCI translation tables used for performing the external/internal VPI/VCI and internal/external VPI/VCI translations described above. The ATM line module 30B3 is, in turn, connected to line termination module (LTM) 304. It is line termination module (LTM) 30B4 which is connected to the physical lines 60.
Examples of the components of a node entity 30 are described, for example, in the following United States Patent Applications (all of which are incorporated herein by reference): U.S. Patent Application SN 08/893,507 for "Augmentation of ATM Cell With Buffering Data"; U.S. Patent Application SN 08/893,677 for " Buffering of PointtoPoint and/or PointtoMultipoint ATM Cells"; U.S. Patent Application SN 08/893,479 for "VP/VC LookUp Function"; U.S. Provisional Application Serial No. 60/086,619 for "Asynchronous Transfer Mode Switch." The structure and operation of node 20 is also further understood with reference to U.S. Patent Application SN
09/188,101, "Asynchronous Transfer Mode Switch," and U.S. Patent Application SN 09/188,265, "Asynchronous Transfer Mode Switch", both of which are incorporated herein by reference.
In the representative node 20 herein described it is node main processor 40 (see Fig. 31) which performs the connection admission control (CAC) procedures of the present invention. In this regard, references hereinafter to an algorithm refer to connection admission control (CAC) procedures executed by node main processor 40. It should be understood that the connection admission control (CAC) procedures of the present invention could be performed for a telecommunications node by a processor located other than on a device board, or by structures other than a processor (e.g., a hardwired circuit). Moreover, although the representative node 20 happens to be an ATMbased node, the connection admission control (CAC) techniques of the present invention are not confined to ATM or to any particular type of telecommunications type (e.g., CDMA). Rather, the connection admission control (CAC) techniques of the present invention can be employed in or for any telecommunications node where traffic management is an issue.
2 Approximations to Bandwidth, Queue Length and Loss Ratio
Consider a general single server queue. A(t) denotes the amount of work which arrives in the interval [t, 0) and S(t) the amount served in the same interval. If more work arrived than can be processed, the surplus waits in the queue, if possible. The workload process is defined by W(0)=0 and W(t) = A(t)  S(t)
and the queue of unprocessed work at time zero is
Q = sup W(t) t ≥ 0
In the foregoing,
A(t) is the amount arriving in [t,0)
R is the mean arrival rate (of stationary arrival process)
S(t) is the amount that can be served in [t,0)
C is the mean service rate
W(t) = A(t)  S(t) is the workload in [t,0) B is the buffer size f_{w}(x;t) is the probability density of W(t)
P_{loss} is the probability of loss
M_{w}(s;t) = E{e^{sW(t)}} Moment generating function of W(t) μ_{w}(s;t) = logM_{w}(s;t) is the log moment generating function of W(t)
If the workload exceeds the buffer size, the excess amount of work, W(t)B, gets lost. The mean amount of work lost in [t,0) is
RtP_{h} t) > (x  B)f_{w}(x; t)dx
There are three causes for the inequality (illustrated in Fig. 1) are that: (1) the queue may be nonempty at t; (2) the queue may be empty in part of [t,0); and (3)the workload may have a maximum within [t,0).
Solving with respect to Pι_{oss}(t) gives
Now the integral is upper bounded by an exponential bound on the factor (xB),
i 0 x < B ke" > \xB x ≥B
where s is a positive parameter and k is a constant chosen such that the curve y = ke^{s} touches the straight line y = x  B. This happens with k =  _{e} ^{~ +sB)} , as illustrated in s
Fig.2. This type of bound is an integrated Chernoff bound.
The upper bound on the average amount of work lost in [t,0) becomes
(x B)f_{w}(x;t)dx ≤
e^{'}^^{sB)}[ef_{w}(x;t)dx≤
e^{~} (1^{+}+^{s}S^{B)} r e^{sx}f_{w}(x;t)dx =
e^{(l+iB)}M_{lv}(s;t) = s μ_{w}(s,t)!,BXXo%(s)
where M_{w}(s;t) = E{e^{sW(t)}} is the moment generating function of f_{w}(x;t) and μ_{w}(s;t) = log_{w}(s;t) is the log moment generating function of f_{w}(x;t). The bound applies with equality if and only if f_{w}(x;t) consists of a single deltaimpulse at x=B+l/s, where the exponential curve touches the line y = x  B.
The bound is tightest for the s that minimizes μ_{w}(s;t)  sB  1  log s. The s that gives the tightest bound must satisfy
δ 1
—— j μι_{ww}((ss;;tt)) BB = 0 ds s An s satisfying this equation does indeed always give a minimum. To prove this, show that the second order derivative is positive,
# 1
 μ_{w}(s; t) + ^{~}2 > °
The proof is outlined using a probability shift argument. The function f_{w}(x;t)e^{sx_} "^{(5;,)} is a probability density function, because it is nonnegative and its integral over all x is 1. A random variable with this probability density has mean c9 c
— μμ_{ww} ((ss;; tt)) aanndd vvaarriiaannccee — — ,μ_{w}(s; t) . Since a variance is nonnegative, this concludes σs σs the proof.
The lower bound on P_{loss}(t) and the upper bound on the integral can be combined into the following approximation:
where q = μ_{w} (s t)  Bs  1  log(Rst)
and B = — <? μ_{w}(s; t)   ^{l} σs s
for all s,t > 0
The maximum of q over all time interval lengths t is used as an approximation to the probability of loss. Assuming derivatives exist, a necessary condition for maximum is dq/dt = 0. Differentiating:
( d 1 ) ( d I] dq = sdB + I — μ_{w}(s; t)  B  Jds + \ τ μ_{w}(s; t)  Jdt
Keeping the buffer limit constant, dB = 0, and using the minimizing s, we get the necessary conditions for maximum
Using the s and t found above, it is found that
dq d^{~}B ^{^ ~S}
Thus, s is the slope of q versus B.
There is no general guarantee that dq/dt = 0 gives a global maximum. Periodic arrival processes give several local maxima and minima.
A necessary condition that the solution found above is a local maximum rather than a minimum is d^{2}q/dt^{2} < 0.
This condition has not had any practical significance in the examples tested.
The solutions it excluded did not admit the traffic anyway for other reasons.
A necessary condition that a local minimum is also global is that the queue is stable, that is, the mean arrival rate is less than or equal to the mean service rate, R < C. Unfortunately, this simple condition excludes the results with constant arrival rate and constant service rate for all s,t > 0, although these results are exact, see Example 1 in Section 5.1 hereof pertaining to constant arrival rates. These results are a buffer size of B = 0, and an excess arrival rate of l/(st), which gets lost. To allow for this case, the weaker condition is used:
R  C < Re^{9} This condition is important in conjunction with periodic arrival processes. They give local maxima in every period, even under heavy overload conditions; but the condition shows that the maxima can not be global.
The results are summarized as follows:
log P_{loss} * q where q = μ_{w}(s t)  Bs  1  log(Rst)
and n ^ / 1
^{B} = — Mw(s\ t) ^{~}  (1) σs s
and = — μ_{w} (s\ t)   at t for s, t > 0 and R  C ≤ Re^{?}
2.1 Constant Service Rate
Assume that the arrival and service processes are independent, and that the service process has a constant rate, C. This gives
μ_{w}(s;t) = μ_{A}(s;t) + μ_{s}(s;t) = μ_{A}(s;t)  Cst
q and its differential become
q = μ_{A}(s;t)  Bs  Cst  1  log(Rst)
, ( dq = sdB  stdC + [ — μ_{A} (s; t)  B  Ct   \ds + I — μ_{A} (s; t) Cs
7 *
Again, to minimize q for constant B, C, and t, the coefficient of ds must be 0. To maximize q for constant B, C, and the minimizing s, the coefficient of dt must be 0. The s and t determined like this have the following interpretation:
s is the slope of q versus B for constant C st is the slope of q versus C for constant B
The result for constant service rate C is
log P_{hss} = q where 9 = A (s; t)  Cst  Bs  1  \og(Rst)
and B r. = — d μ_{A} ( ,s; t s).  C „t   1
(2) as s Ϋ and c = I ( <9 — μ_{A} (s; t)   s ^ t tJ for s, t > 0 and R  C < Re^{?}
2.2 General Service
Solving equations (1) for (s,t) is difficult. Instead, we shall use (1) with the actual arrivals and service replaced by something else.
The "unused typical arrivals" problem formulation in section 2.2.1 presents a solution in a form suitable for implementation in a realtime CAC algorithm. It's suitability comes from a "single point calculation" property.
The problem formulations in sections 2.2.22.2.4 are special cases and variants of this. Despite the solutions look simpler they are less suitable for implementation in a realtime CAC algorithm. This is because they do not have the "single point calculation" property, but rather a "leaning banana" trouble.
2.2.1 Unused Typical Arrivals Formulation
Replace the actual arrivals A(t) by A(t) + U(t), where U(t) is unused typical arrival traffic.
Generate U(t) as follows: Choose a typical set of connections, and call its mean arrival rate RQ and its log moment generating function μ_{0}(s;t). Add independent connections in the same proportions as in the typical set, but m times as many, to form U(t) with mean arrival rate mRo and log moment generating function μu(s;t) = mμ_{0}(s;t). Choose m such that the traffic satisfies equations (1).
The result for general service S(t) with unused typical arrivals is
l°g Pioss ~ g
where q  Mw (si ^{f})  ^{s} — Mw tø 0 + μ_{0} (s; t)  s — μ_{0} (s; t)  log((R + mR_{Q} )st) σs V as J
and B = — μ_{w} (s; t) + m — μ_{0} (s; t)   (3) σs s s , . 1 — μ_{w} (s; t)   and m =  ≤^ 
— o(s; t) dt for s,t>0 and R + mR_{0}  C ≤ (R + mR_{0})e'^{1}"'"
It should be noted that m=0 corresponds to the maximum admissible load of real traffic. If m is nonnegative, the service S(t) is sufficient, otherwise S(t) is insufficient. B and q in (3) are not the actual buffer size and log loss probability, but the buffer size and log loss probability that would arise if the unused typical arrivals were added to the l o actual arrivals.
The terms m and B in (3) are linear functions of the derivatives of μ_{w}(s;t). q is the sum of a linear and a logarithmic term.,
q = z  log c d d where z = μ_{w} (s; t)  s — μ_{w} (s; t) + m\^μ_{0} (s; t)  s  μ_{0} (s; t) j  log(st) (4)
and c = R + mR_{0}
A CAC algorithm can keep track of c, z, B, and m by adding contributions from 15 new connections and subtracting contributions from connections being cleared. The "unused typical arrivals" formulation of the problem has a "single point calculation" advantage. If the real arrivals come from a typical traffic mix, then the mean rate and log moment generating function of A(t)+U(t) do not vary with the real traffic load. This means that a point (s,t) that admits the traffic at the maximum admissible load also admits at all lower loads. If the real traffic deviates from the typical mix the sum of real and imaginary traffic may still be sufficiently similar to the typical mix to allow a single point (s,t) to admit over a range of real traffic mixes and loads.
2.2.2 Unused Arrival Bandwidth Formulation
Replace the actual arrivals A(t) by A(t)+Ut, where U is the unused arrival bandwidth.
The result for general service S(t) with unused arrival bandwidth U is
where q = w(^{s'}> t) + Ust  Bs  l  log((R + U)st)
and B r> = — & μ_{w} ( ,s; t N) + U T 7t   1
(5) as s
and V =   \ — μ_{w}(s; t)   s K t t^{)} for s, t > 0 and R + U  C ≤ (R + U)e^{q}
If U is nonnegative, S(t) is sufficient, otherwise S(t) is insufficient. B and q in Equation Block (5) are not the actual buffer size and log loss probability, but the buffer size and log loss probability that would arise if the actual arrivals were increased by the unused bandwidth.
The log loss probability q in Equation Block (5) is the sum of a linear and a logarithmic term, q = z  log(R + U) where z = μ_{w} (s; t) + Ust  Bs  I  l g(st)
An admission control algorithm can keep track of B, R, U, and z by adding contributions from new connections and subtracting contributions from connections being cleared.
A drawback of using Equation Block (5) for realtime admission control is that the (s,t)region admitting the last connection does not always overlap the (s,t)region admitting the first connection. In (s,t,load)space the region admitting the loads may look like a "leaning banana". Thus, it is not sufficient just to calculate Equation Block (5) in a single point (s,t).
2.2.3 Unused Service Bandwidth Formulation
Replace the actual service S(t) by S(t)Ut, where U is the unused service bandwidth.
This gives the same workload on the queue as in the unused arrival bandwidth formulation. U and B are the same as before, but q is greater when U>0. At full load, when U=0, the two unused bandwidth formulations give the same result.
The result for general service S(t) with unused service bandwidth U is
where q = w (s; t) + Ust  Bs  1  log(Rst)
and B = — μ_{w}(s; t) + Ut   (6) as s
for s, t > 0 and R  (C  U) ≤ Re^{?m»} If U is nonnegative, S(t) is sufficient, otherwise S(t) is insufficient. B and q in (6) are not the actual buffer size and log loss probability, but the buffer size and log loss probability that would arise if the actual service was reduced by the unused bandwidth.
The set of points (s,t) admitting the arrivals differs between the two unused bandwidth formulations. If Equation Block (6) admits the arrivals at (s,t), then so does Equation Block (5); but the converse is not true. If Equation Block (5) admits the arrivals at (s,t), the Equation Block (6) may admit or be uncertain.
The log loss probability q in Equation Block (6) is the sum of a linear and a logarithm term,
q = z  log R where z = μ_{w} (s; t) + Ust  Bs  I  log(st)
B, R, U, and z are all linear functions of the log moment generating function of the workload W(t). An admission control algorithm can keep track of B, R, U, and z by adding contributions from new connections and subtracting contributions from connections being cleared.
The unused service bandwidth formulation suffers from the same "leaning banana" trouble as does the unused arrival bandwidth formulation.
2.2.4 Arrivals Multiplier Formulation
Imagine increasing the arrivals k times by adding independent connections from the same traffic mix as the actual connections. The resulting arrivals have mean rate and log moment generating function
R_{k} = kR ω te = ^{k}M_{A} (^{s'}^ Select the multiplier k such that the workload resulting from imaginary arrivals and actual service satisfies Equation Block (1). The result is
log P_{loss} * q where q = kμ_{A} (s; t) + / , (.?; t)  Bs  I  log(kRst) d , . d 1 and B = k — μ_{A} (s; t) + — μ_{s} (s; t)   (7) as as s
— <? μ_{s}( ,s t Λ)   and at for s, t > 0 and kR  C ≤ k Re^{q}
If k> 1 , B<B_{0}, q<q_{0} at some (s,t), the connections are admissible. If k<l, B>B_{0}, 5 q>q_{0} at some (s,t), the connections are inadmissible.
The admission control algorithm of the present invention keeps track of four σ state variables R, μ_{A} (s; t), — μ_{A} (s; t), — μ_{A} (s; t) by adding contributions from new connections. The algorithm can calculate q, B, and k from these state variables.
The arrivals multiplier formulation suffers from the same problem as the unused l o bandwidth formulation, that the point (s,t) that admits the last connection at maximum load will in general not admit the first connection.
2.3 Asymptotic Expressions for Large t
Assume that the workload process W is stationary and ergodic. Assume that it satisfies the conditions of the GartnerEllis theorem; that is, assume that the asymptotic 15 log moment generating function
exists and is finite for all real s, and that h_{w} is differentiable. That h_{w} is convex, positive and increasing for s>0 can be directly verified. This means that for large t the following approximation can be used:
μ_{w}(s; t) = h_{w}(s)t
For constant rate and Poisson process, this product form of μ_{w}(s;t) holds for all t.
For other interesting processes, e.g. leaky bucket shaped traffic, it is poor for small t. If μ_{A}(s:t)=h_{A}(s)t and the service process has constant rate C, the results becomes
log P_{l0ii} ~ q where q = h_{A} (s)t  Cst  Bs  I  \og(Rst)
and B = h_{A} ' (s)t  Ct  
and C =  \ h_{A} (s)   s tJ l_ for s, t > 0 R ≤ C + st
Delay
3.1 Constant Service Rate
For a given pair (s,t), Equation Block 3 gives a buffer size B and the logarithm q of the probability of exceeding B. Data arriving to a B long queue served at constant rate C will leave the queue after a delay of D=B/C. If the imaginary traffic is positive, the delay of the real traffic will be smaller. Hence, the (le^{q}) fractile of the delay is at most B/C, and
P(D > ^) ≤ e" (8) where the equality sign applies when the system is fully loaded.
4 Multiple Queues
4.1 Multiple Priorities, Single Server
Consider a system consisting of two queues, H and L, served by a single server as shown in Fig. 3. The server serves queue H whenever that queue is not empty. The server serves queue L when queue H is empty and queue L is not. Queue H sees the full service of the server. Queues H and L together also see the full service of the server. Queue L sees what queue H leaves unused. The log moment generating function of the service offered to queue L is determined as follows.
The workload on queue H is W_{H}(t) = A_{H}(t)S(t). The service offered to queue L is
S_{L} (t) = (S(t)  A_{H}(t))^{+} = W^{~} _{H} (t)
W_{H} (t) for W_{H} (t) < 0 where W _{H}(t) 0 for W_{H} (t) > 0
From this, the log moment generating function of the service offered to queue L is determined:
μ_{St} (s; t) = μ_{rH} (~s; t)
This is a function of μ_{AH} (s; t) , but unfortunately, a nonlinear one. This means that contributions can not just be added to it from every highpriority connection. A practical approximation is to replace W_{H} ^{"} by W_{H} and use
M_{sL} (s; t) ~ μ_{WH} (~s; t) (9)
This is equivalent to loading queue L by the total traffic A and offering it the full service S. It is a good approximation if the probability of positive workload on queue H is small. 4.1.1 Admission Control for Multiple Priorities, Single Server, Shared Buffer
In this scenario, queues H and L share a common buffer of limited size. Equation Block (3) is used with workload W_{H} = A_{H}  S to evaluate q_{H} at a pair (s,t).
If q_{H} is admissible, use Equation Block (3) with workload W = A  S. Evaluate the overall variables m, B, and q at (s,t). Find the lowpriority log loss probability as the solution of the following loss rate equation.
Re^{?} = R_{H}e^{q}" + R_{L}e"^{L}
Re^{9}  R e^{q}" If q_{L}≠0, the solution is q_{L} = log . To prevent division by 0, check
^{R}L that
Re'  R^e*" < R_{L}e"^{L}" (10)
instead of checking q_{L}≤qi_{maX} directly.
Show that all of m, B, q_{H}, and q_{L} are admissible or inadmissible at (s,t), if possible.
There may be more than two priority levels. Evaluate q_{L} of the each lower level as above using the q of the total traffic on this and all higher levels. At the lowest level, show that m, B, q_{H}, and all q_{L} are admissible or inadmissible at (s,t), if possible.
4.1.2 Admission Control for Multiple Priorities, Single Server, Individual Buffers
In this scenario, queue H has a buffer of limited size, and queue L has a buffer of limited size. Equation Block (3) is used with workload W_{H}=A_{H}S. Show that all of the high priority variables m_{H}, B_{H}, and q_{H} are admissible or inadmissible at a pair (s_{H},t_{H}), if possible. If m_{H}, B_{H}, and q_{H} are admissible, use Equation Block (3) with workload W_{L}=A_{L} S_{L}, approximating S_{L} as in Equation Block (6). Show that all of the lowpriority variables m_{L}, B_{L}, and q_{L} are admissible or inadmissible at a pair (s_{L},t_{L}), if possible.
There may be more than two priority levels. Repeat the evaluation for layer L for each lower priority level using the sum of traffic on all higher layers for A_{H}.
4.2 Single Priority
Consider a system consisting of N queues served by a single service as illustrated in Fig. 4. The server serves the nonempty queues according to a schedule giving them an equal share of the service. This schedule could be round robin or according to a snapshot of queue states taken when a previous snapshot has been processed. The N queues together see the full service of the server. An individual queue sees what the other queues leave unused. The situation looks like the one for a lowpriority queue seeing the service the highpriority queues leave unused, and we shall use the same approximation (Equation 9), restated here for convenience.
μ_{Sι} (s; t) « μ_{WH} (s; t) (9)
Here S_{L} is the service seen by an individual queue, and W_{H} is the total workload on the other queues. Again, this approximation gives the same distribution of the length of one queue as the distribution of the sum of the lengths of all the queues. Fig. 5 shows a simulation result justifying such approximation.
Fig. 5 shows distributions of queue lengths. The system comprised 21 queues of equal, low, priority served by a single server. Equal traffic load on all queues with exponentially distributed time between bursts, geometrically distributed number of arrivals in a burst, and constant interarrival time within a burst. Scheduling according to a snapshot of queue states, queue 0 served first and queue 20 last. In Fig. 5, the lower set of curves shows the length of queue 0, which was served first in the snapshot; the middle set of curves shows the length of queue 20, which was served last in the snapshot; and the top set of curves shows the sum of the lengths of all 21 queues. 4.2.1 Admission control for Single Priority, Single Server, Shared Buffer
In this scenario, the N queues share a common buffer of limited size. This case is equivalent to the single queue case. Equation Block (3) is used with workload W=A S. Evaluate the overall variables m, B, and q at a pair (s,t). Show that all of U, B and q are admissible or inadmissible at (s,t), if possible.
4.2.2 Admission Control for Single priority, Single Server, Individual Buffers
In this scenario each of the N queues has a buffer of limited size. Let A_{L} be the arrivals to the queue considered, and let A_{H} be the sum of arrivals to the other queues. Equation Block (3) is used with service S_{L}=SA_{H}. Show that the variables m_{L}, B_{L}, and q_{L} of the queue considered are all admissible or inadmissible at a pair (s_{L},t_{L}), if possible.
4.3 Queues with Individual Service and Shared Buffer
The sum of workloads on queues with independent servers does not give a correct estimate of the sum of queue sizes. Fig. 6 shows an example illustrating this point. In Fig. 6, queue 2 is unstable and grows beyond all limits, while the sum of workloads on the queues is less than the sum of services. It is postulated that the total buffer needed for queues with independent servers is approximately equal to the need of the longest queue. A motivation is that the queues are independent and a long queue has a small probability, so when one queue is long, the others are likely to be around their mean length which is much smaller.
4.4 Queues in the SPAS Switch
Fig. 7 shows a simplified diagram of the queueing system in the spatial switch SPAS (core 24 in Fig. 31) in a wideband CDMA telecommunications system. As mentioned above, details of the spatial switch SPAS beyond those present here are discussed in U.S. Patent Application SN 09/188,101 , "Asynchronous Transfer Mode Switch," and U.S. Patent Application SN 09/188,265, "Asynchronous Transfer Mode Switch", both of which are incorporated herein by reference. The switch core 24 comprises rows for incoming data and columns for outgoing data. In the crosspoints between rows and columns there are small buffers which are omitted in Fig. 7. The space switch interface modules SPIM contain ingress and egress queues towards and from the switch core, respectively.
The ingress queues in a SPIM are organized as a queue per egress SPIM and ingress priority. The idea is to store incoming cells in ingress queues and transfer them in an ingress priority order towards the switch core when the corresponding crosspoint is free. Ingress queue (i, j, p) is located in SPIM i and carries traffic towards SPIM j on priority level p.
The ingress queues in a SPIM share a common buffer memory. Ingress queues in different SPIMs do not share a buffer.
The reading of cells from the ingress queues follows the following rules:
• Cells with the highest priority will be fetched until the queues are empty or blocked by occupied crosspoints.
• Lower priority cells will be read as soon as there are no cells in queues with higher priority or that all of the higher priority cells are blocked by occupied crosspoints.
A roundrobin mechanism is used to give equal bandwidth to the different ingress queues within the same priority level.
The switch core has two priority levels. To get a high priority cell through the switch core when a low priority cell blocks the crosspoint, a command is sent as a special management cell called Plus Priority cell. This cell is terminated in the core and any cell in the crosspoint buffer gets its priority increased. In this way, the crosspoint will be emptied within a predetermined time.
The egress SPIM serves the crosspoints in its column according to a snapshot mechanism. When the SPIM has processed a previous snapshot it takes a new snapshot of the crosspoint buffer states. The SPIM empties the nonempty buffers in the snapshot in increasing number order.
The egress queues are organized as a queue per egress priority and outgoing branch from the SPIM. The idea is to store incoming cells in queues and transfer them in egress priority order. The egress queues in a SPIM share a common buffer memory. This memory is separate from the ingress buffer memory.
4.4.1 Ingress Admission Control
This section describes how to use the "unused typical arrivals" results for admission control of the ingress queues.
Row i and column j in the switch core both limit the service of the ingress queues in SPIM i towards SPIM j. The ingress admission control algorithm checks that both of these potential bottlenecks give sufficient service.
The ingress queues served by a row share a buffer in a SPIM. Hence it suffices to keep track of the total workload per priority level when checking for row service.
The ingress queues served by a column reside in different SPIMs, and therefore do not share a buffer. Hence one must keep track on workload on individual queues when checking for column service. Queues sharing a buffer in a SPIM do not share column service.
The ingress admission control algorithm for row service uses the following state variables for some set of points (s;t),
c_{R}(p,i) Average rate of real + imaginary traffic on priority levels p and higher from SPIM i
z_{R}(p,i,s;t) Linear term in long loss probability of traffic on priority levels p and higher from SPIM i
B_{R}(p,i,s;t) Buffer size of traffic on priority levels p and higher from
SPIM i m_{R}(p,i,s;t) Multiplier of unused typical traffic on priority levels p and higher on row i
The ingress admission control algorithm for column service uses the following state variables for some set of points (s;t),
c_{c}(p,i,j) Average rate of real + imaginary traffic on priority levels p and higher from SPIM i towards SPIM j
z_{c}(p,i,j,s;t) Linear term in log loss probability of traffic on priority levels p and higher from SPIM i towards SPIM j
B_{c}(p,i,j,s;t) Buffer size of traffic on priority levels p and higher from SPIM i towards SPIM j
m_{c}(p,j,s;t) Multiplier of unused typical traffic on priority levels p and higher on column j
The typical traffic has the following characteristics for some set of points (s;t),
Ro(p,i) Average rate of typical traffic on priority levels p and higher from SPIM i
μ_{0}(p,i,s;t) Log moment generating function of typical traffic on priority levels p and higher from SPIM i
— μΛp, i, s; t) Partial derivative with respect to s, typical traffic on priority σs levels p and higher from SPIM i
— μ_{0} (p, i, s; t) Partial derivative with respect to t, typical traffic on priority levels p and higher from SPIM i
For shortness, the coefficients of —   are introduced: t a_{R}(p,i,s;t) = in the expression for R
d μ_{0}(p, i, s;t) s^{~} μ,(p, i, s; t) a_{z}(p,i,s;t) = in the expression for z d_ μ_{0}(p,i,s;t) dt σ —_{s}μ_{0}(p,ι,s;t) ι_{B}(p,i,s;t) =  in the expression for B μ_{0}(p,i,s;t) dta_{m}(p,i,s;t) = in the expression for m d_ μ_{0}(p,i,s;t) dt
The initial values for row service, with no traffic through the queues, are
c_{R}(p,i,s;t) = α_{R} (p, i, s; + 
1, z_{R}(p,i,s;t) = α_{z}(p,i,s;t) C.s +  \og(st)
V
(11) f Λ
B_{R}(p,i,s;t) = C,t + α_{B}(p,i,s;t) Cs +  v ' tJ f m_{R}(p,i,s;t) = α_{m}(p,i,s;t) Cs +  v t
The initial values for column service, with no traffic through the queues, are ( 1 c_{c}(p,ι,j,s;t) = a_{R}(p,ι,s;t) Cs +  t
z_{c}(p,i,j,s;t) = a_{z}(p,i,s;t) ( C s + I  log(^t) t
(12)
B_{c}(p,i,j,s;t) = C_{j}t + a_{B}(p,ι,s;t) ( C + lλ
V ^{J} tJ
m_{c}(p,i,j,s;t) = a_{m}(p,i,s;t) C.s + tJ
Input to the ingress admission control algorithm about a new connection is
p_{a} Priority level
i Ingress SPIM
j Egress SPIM
r Average rate
μ_{a}(s;t) Log moment generating function of arrivals at a set of points (s;t)
d_
M_{a}(s;t) Partial derivative with respect to s ds
d_
M_{a}(s;t) Partial derivative with respect to t dt
The new connection adds the following contributions to the row state variables,
The new connection adds the same contributions to the affected column state variables,
d Ac_{c}(p, k, j, s; t) = r  a_{R} (p, i, s; t) — μ_{a} (s; t)
d d
Az_{c} (p, k, j, s; t) = μ_{a} (s; t)  s — μ_{a} (s; t)  a_{z} (p, i, s; t) — μ_{a} (s; t) ds d d d
AB_{c}(p, k, j, s; t) = — μ_{a} (s; t)  a_{B}(p, i, s; t) — μ_{a} (s; t) (14)
d Am_{c}(p, k, j, s; t) = a_{m}(p, i, s; t) — μ_{a} (s; t)
for p ≥ p_{a} , all k, all (s; t)
4.4.2 Egress Queues
An outgoing link serves its egress queues at a constant rate. A problem is the traffic arriving to the egress queues. This traffic has changed by passing through the ingress queues and the switch core. As an approximation, assume that the traffic has not changed. This situation is equivalent to handling ingress queues with row service only, see above.
4.5 The Admission Control Algorithm
The algorithm initializes the state variables c, z, B, and m, as described in Equation Blocks (11)(12). In order to admit a new connection, the algorithm checks every affected priority level. If all priority levels admit the connection, it is admissible, otherwise not.
In order to check a queue, the algorithm checks it over one or more time intervals t. If no t rejects the connection, and some t admits it, the queue admits the connection over the time interval t, otherwise the queue rejects the connection.
In order to check a queue over a time interval t, the algorithm checks it over one or more values of s for the given t. If some point (s;t) rejects the connection, the time interval rejects it. If no point (s;t) rejects, and some point (s;t) admits the connection, the time interval t admits it.
In order to check a queue at a point (s;t) the algorithm increments the state variables c, z, B, and m, as described in equations (13)(14). It checks the rate condition last in equations (3). If the rate condition is OK, the algorithm calculates q as described in equation (4). It corrects the loss probability for losses on higher priority levels at the same point as described in equation (10). If the corrected loss probability is sufficiently small, and the buffer B is sufficiently small, and the traffic multiplier m is nonnegative, and the rate condition is OK, then the queue admits the connection at point (s;t). If the corrected loss probability is negative, then the queue rejects the connection at point (s;t). Otherwise the queue is indecisive at point (s;t).
If the algorithm admits a new connection, it updates all affected c, z, B, and m. If the algorithm rejects a new connection, it reverts to the previous values of all affected c, z, B, and m.
In order to release an established connection, the algorithm decrements all affected c, z, B, and m by the amounts it added when it admitted the connection.
5. Examples of Log Moment Generating Functions
5.1 Example 1. Constant Arrival Rates
The workload in an interval of length t is A(t) = Rt
where R is the arrival rate. The log moment generating function (illustrated in Fig. 8A, for constant rate source with R=l) and its derivative with respect to s (illustrated in Fig. 8B) and its derivative with respect to t (illustrated in Fig. 8C) are
μ_{A} (s; t) = log f e^{sx}δ(x  Rt)dx = sRt
— μ_{A} (s; t) = sR dt
d ^{2} μ_{A} (s; t) = R dsdt
μ_{A} (s; t) = 0 t^{2}
In this case, the asymptotic approximation is exact:
μ_{A}(s;t) = h_{A}(s)t where h_{A}(s) = sR
The approximation with constant service rate C become
log P_{loss} * q where q =  \og(Rst) and B = 0
and C = R  —
St
for s, t > 0 and — < Re^{9}™"^{1} st
This is the case where the bound applies with equality, since f_{w}(x; ) consists of a single deltaimpulse at x = (RC)t = B+l/s, where the exponential curve touches the line y = x  B. In this case, the approximation is exact.
5.2 Example 2. ONOFF Periodic Fluid Source
The source is ON for T_{on} and OFF for T_{off}. The period is T_{on} + T_{off} = T. In the
T ON state, the source generates data at the peak rate R — , and in the OFF state, it on generates no data. This is an extreme behavior acceptable by a leaky bucket regulator with mean rate limit R, and bucket size RT_{off}. The phase of this periodic pattern is uniformly distributed in [0,T).
Fig. 9A  Fig. 9E are graphs showing ONOFF periodic arrivals. In Fig. 9A  Fig. 9E, A(τJ+τ) is the amount arriving in [τ,t+τ). The phase τ is uniformly distributed in [0, T_{on} + T_{off}). From Fig. 9A  Fig. 9E, it can be seen that the density function of A is the sum of two deltaimpulses and a uniform distribution between them. δ(x  nRT) for t = nT f_{A} = ' aδ(x  x_{t} ) + bU(x; x_{λ} , x_{2}) + cδ(x  x_{2}) otherwise where
For t equal to an integer number of periods, this degenerates to a single delta impulse. For t equal to an integer plus one half number of periods and T_{on} = T_{off}, this degenerates to a uniform distribution. Table 1 lists the parameters and their partial derivatives for 0<t<T.
TABLE 1
ONOFF PERIODIC ARRIVALS, PARAMETER VALUES IN DENSITY
FUNCTION
For t = nT + αT, 0<α<l use
f_{A}(x;nT+αT) = f_{A}(xnRT;αT)
The following auxiliary variables are now introduced:
y = s(x_{2}x,) z = e^{"y} _ 1  z
V y w = az + bv + c
with partial derivatives
dy ds dz dy
= e ds ^{~}ds dzl (1  z) dy dv for y ≠ 0 dsy yl ds ds^{''} ]_dy for y = 0
1 ds dw dz dv — = a — + b— + c ds ds ds
and
dy ( dx_{2} dx dt^{~} Λ dt dt dz _ _ __{y} dy dt^{~ e} dt dzl_ _ (1z) dy for y ≠ 0 dv dty yl dt dt^{' } ^{1} dy ^{γ} for v = 0
2 dt ^{y} dw da dz db , dv dc
— = — z + a — + — v + b — + — dt dt dt dt dt dt
The log moment generating function and its derivatives can now be written as
μ_{A}(s;nT + aT) = s(x_{2} + nRT) + log w d 1 dw
— μ_{A} (s; nT + aT) = x_{2} + nRT + ds w ds d dx_{2} 1 dw
— μ, (s; nl + al) = s —  + dt ^{A} dt w dt
Fig.10A is a graph showing the Log moment generating function, and Fig.10B and Fig. IOC its derivatives with respect to s and t, respectively, for ONOFF periodic source with R=l, T=l, T_{on}=0.2. 5.3 Example 3. Discrete Periodic Arrivals
An amount of RT arrives at times t = nT + αT, where n is integer and αe [0,1) is constant. The phase αT is uniformly distributed in [0,T).
In an interval of length t  nT + αT arrives an amount A(t) of either (n+l)RT or nRT,
P(A(nT + αT) = nRT) = 1α P(A(nT + αT) = (n+l)RT) = α
The log moment generating function, its derivatives, and the asymptotic log moment generating functions are
μ_{A} (s; nT + aT) = sR(n + 1)7 + log((l  a)e^{'sRT} + a
— ^{d} μ_{A t}(s; nT τ + aT _{T}) = R r_{>}((n  +_{.} n 1)7r RT( —l  a)e^{~iRT} d (1  a)e^{~} s^{s}R^{κ}T' + a d_ sRT μ_{A} (s; nT + aT) = — dt 7 (1  a)e^{iKI} + a a^{1} aR^{2}T^{2} (l  a)e sRT
, μ_{A} (s; nT + aT) ds^{2}^^{A} ' ' ((1  a)e^{~sRT} + a)^{2}
—  μ_{A} (s; nT + aT) = — —  — — dt s ((1  a)e + a)
<? μ_{A} ( ,s; . nT T + , aT) (  ^{e~sRT}y d ^{A} ' T^{2}((l  a)e^{'sRT} + a)^{2} h_{A} (s) = sR
Notice that the asymptotic log moment generating function is the same as for a constant rate process. Using this, the asymptotic log moment generating function in the approximations is thus equivalent to using a constant rate model of the arrival process.
Fig. 11 A is a graph showing the log moment generating function, and Fig. 1 IB and Fig. 1 IC are graphs showing the derivatives of the log moment generation function of Fig. 11 A with respect to s and t, respectively, for discrete periodic source with T=l, R=l.
5.4 Example 4. Poisson Arrivals
The mean interval time is T. Each arrival contributes an amount of RT. The probability of k arrivals in time t is (e^{_t/T}(t/T)^{k})/k!. The log moment generating function and its derivatives of the arrival process are (as explained in A. Papoulis, "Probability, Random Values, and Stochastic Processes", McGrawHill, 1965),
μ_{A} (s; t) =  (e'^{κl}  1)
M_{A} (s; t) = Rte sRT ds
ds^{2} d^{2} μ_{A} (s; t) = Re sRT dsdt
Also in this case, the asymptotic approximation is exact: μ_{A}(s;t) = h_{A}(s)t where h_{A}(s) = (e^{sRT}l)/T
Fig. 12A is a graph showing the log moment generating function, and Fig. 12B and Fig. 12C are graphs showing the derivatives of the log moment generation function of Fig. 12A with respect to s and t, respectively, for a Poisson source with T=l, R=l . 5.5 Example 5. Markov Fluid Arrival Process
A source is called a Markov fluid if its timederivative is a function of a continuoustime Markov chain on a finite state space. Let 1, ..., m be the state space and
Q_{A} = [q,_{j}] the irreducible transition rate matrix
q,_{j} the transition rate from state i to another state j
q„ =  X q,_{j} minus the sum of transition rates from state i
π = (π , , ... , π_{m}) the row vector of steadystate probabilities, is the solution of πQ_{A} = 0 and πl = 1
R_{A} = diag(R, , ... , R_{m}) diagonal matrix of arrival rates
R, arrival rate in state i
1 a column of l 's
λ(X) The largest real eigenvalue of a matrix X
The log moment generating function (see Kesidis et al, "Effective Bandwidths for Multiclass Markov Fluids and Other ATM Sources", IEEE Trans. Networking, Vol. 1, No. 4, pp. 424428, Aug. 1993), and its partial derivatives and asymptotic log moment generating function are μ_{A}(s;t) = log ^{ρ}^^{)}'l)
_d ^{'} (Q_{A}+sR_{A})t j
Mxtot) —<? , i ds π έ^{Q<} sR_{λ})tt π ( ds
^{(} 1
— ^{d} μ_{A}(s t) = dt π< iQΛ+^{S}*Λ)' π(Q_{A} +sR_{A})e (QΛ+*XΛ)I 
h_{A}(s) = λ(Q_{A} +sR_{A})
The partial derivative with respect to s is not simple in general. To find it, diagonalize
Q_{A}+sR_{A} = N(s)D(s)V^{1}(s) where D(s) is a diagonal matrix of eigenvalues of Q_{A}+sR_{A} and V(s) is a matrix of column eigenvectors
Now use
and find after some calculation
d _{6}(QA+SR_{A})I ds
5.5.1 HIGH/LOW Rate Source
A HIGH/LOW rate source is a special case with two states, QA = (transitions / s)
and an ON/OFF source is special case of this with R_{2}=0.
The steady state probabilities of the source are
Direct calculation gives the eigenvalues
h_{A} (s) = , = — l a + Vα^{2}  4/3 ]
λ_{2} = I a  4a^{2}  4b]
where α = Q_{X 2} + Q_{2 ]}  s(R_{2} + R, ) and b = s^{2}R_{2}R,  s(Q, _{2}R_{2} + Q_{2} ,R, )
Diagonal eigenvalues and column eigenvectors matrices are
Derivatives with respect to s are
where
da and where R_{2}  R, ds db and = lsR_{x}R_{2}  Q, _{2}R_{2}  Q_{2} , R ds
Fig. 13A is a graph showing the log moment generating function, and Fig. 13B and Fig. 13C are graphs showing the derivatives of the log moment generation function of Fig. 13A with respect to s and t, respectively, for an ON/OFF Markov fluid source with T=1/Q +1/Q_{2>1}=1 , T_{on}=l/Q,_{,2}=0.2, R=l .
5.6 Always ON or OFF Fluid Arrival Process
This is a limiting case of a process with very heavy tails on the holding time distributions. The source is ON always with probability P_{ON}, generating data at rate R/P_{ON}, or it is OFF always with probability P_{OFF}=lP_{ON}, generating no data. The probability density function of the amount of data received in time t is
f_{A}(x; t) = P_{OFF}δ(x) + P_{0N}δ(x    t) The log moment generating function and its derivatives are
d Rt μ_{A}(s;t) ds R
s J
P ^{1} OFF e^ + P O,N d_ sR μ_{A}(^{s}\t) = dt
P_{n} OFF' + p, ON sR
OFF R D ^{2z}t * ^{22}e , p O,N μ_{A}(^{s'}>t) ds^{2} sRl
P ^{1} OFF e^ + P ON
sRl sRt
P_{n} OF_{m}Fe' ^{P}°^{»} + „ ON„ +sRt^^e ^{p}°" μ_{A}(s;t) = R ON δsdi _ sRl
^{1} P OFF e ^{PofJ} + ^ P ^{x} ON
Fig.14A is a graph showing the log moment generating function, and Fig.14B and Fig.14C are graphs showing the derivatives of the log moment generation function of Fig.14A with respect to s and t, respectively, for always ON or OFF fluid source with P_{0N}= 0.2 and R= 1. 6 Examples of Loss, Buffer, and Bandwidth Consumption
This section defines some traffic classes spanning a wide range of bandwidth and burstiness. It uses traffic cases spanning a wide range of link loads to illustrate regions of (s,t) admitting or rejecting a traffic case. Table 2 lists traffic classes.
Fig. 15A Fig. 30B show contours of q = In 10^{"8}= 18.4, B =3 Mbit, and m = 0 bit/s, and m = m_{max}, for differing traffic cases and the region(s) bounded by these contours admitting or rejecting the particular traffic case. Fig. 15A Fig. 30B assume a link bandwidth of C_{0} = 150 Mbit/s. The typical traffic used is the actual traffic itself unless otherwise stated.
Table 3 lists numbers traffic cases with one connection.
Fig. 15A illustrates the foregoing one connection traffic case 1; Fig. 15B illustrates one connection traffic case 10. Both Fig. 15A and Fig. 15B show contours of q, B, and m, for one connection, average rate lOkbit/s. Fig. 15A shows case 1 (constant rate); Fig. 15B shows case 10 (always ON or OFF, P_{ON}=0.01). No B>3Mbit is found for m<m_{max}; no m<0 is found for m<m_{max}.
Fig. 16A  Fig. 16D illustrate traffic cases 2, 3, 4, 5, (showing contours of q, B, and m) for one connection, average rate lOkbit/s, average period 10 ms. Fig. 16A shows case 2 (ONOFF periodic, P_{ON}=0.01); Fig. 16B shows case 3 (discrete periodic); Fig. 16C shows case 4 (ONOFF Markov, P_{ON}=0.01); and Fig. 16D shows case 5 (Poisson).
Fig. 17A  Fig. 17B illustrate traffic cases 6, 7, 8, 9 (showing contours of q, B, and m) for one connection, average rate lOkbit/s, average period 10 s. Fig. 17A shows case 6 (ONOFF periodic, P_{ON}=0.01); Fig. 17B shows case 7 (discrete periodic); Fig. 17C shows case 8 (ONOFF Markov, P_{ON}=0.01); and Fig. 17D shows case 9 (Poisson).
Fig. 18A  Fig. 18D illustrate traffic cases 1 1, 12, 13, 14 (showing contours of q, B, and m) for one connection, average rate lOkbit/s, average period 10 ms.. Fig. 18A shows case 11 (ONOFF periodic, P_{ON}=0.01); Fig. 18B shows case 12 (discrete periodic); Fig. 18C shows case 13 (ONOFF Markov, P_{ON}=0.01); Fig. 18D shows case 14 ( Poisson).
Fig. 19A  Fig. 19D illustrate traffic cases 15, 16, 17, 18 (showing contours of q, B, and m) one connection, average rate lOkbit/s, average period 10 s. Fig. 19A shows case 15 (ONOFF periodic, P_{ON}=0.01); Fig. 19B shows case 16 (discrete periodic); Fig. 19C shows case 17 ( ONOFF Markov, P_{ON}=0.01); Fig. 19D shows case 18 (Poisson). Traffic case 16 of Fig. 19B is clearly inadmissible. A burst of 1 Gbit arrives instantaneously to a buffer of size 3Mbit, which looses 99.7% of the arriving data. Yet Equation Block (3) admits the connection at some points (s,t) with t>l period = 100 s. The reason for this is that the workload over long periods looks acceptable, while it is unacceptable over short periods (see also Fig. 1). The global maximum of q occurs in the first period oft. The admitting points found are at local maxima in later periods and thus false solutions. The conclusion of this is that one must choose the working point (s,t) for periodic traffic within the first period. Another peculiarity of traffic case 16, is that no working point (s,t) is able to reject the case. In the region below the contour labeled "18.4" in Fig. 19B, q and B are far from admissible, while m is close to 0; but m>0 everywhere. Fig. 20 illustrates traffic case 19 (showing contours of q, B, and m) for one connection, average rate lOMbit/s. Case 19 is always ON or OFF, P_{ON}=01.
Table 4 lists numbers traffic cases (cases 20  38) with the maximum number of connections from a single traffic class admitted by Equation Block (3).
Table 5 lists numbers traffic cases (cases 39  57) with the minimum number of connections from a single traffic class not admitted by Equation Block (3).
Fig. 21 A  Fig. 2 ID illustrate traffic cases 20, 29, 39, 48 (again showing contours of Contours of q, B, and m), with maximum admissible and minimum inadmissible numbers of connections, average rate lOkbit/s. Fig. 21 A shows case 20 (constant rate, 15000 connections); Fig. 2 IB shows case 29 (always ON or OFF, P_{ON}=0.01, 9508 connections); Fig. 21C shows case 39 (constant rate, 15001 connections); Fig. 2 ID shows case 48 (always ON or OFF, P_{ON}=0,01, 9509 connections).
Fig. 22A  Fig. 22D illustrate traffic cases 21, 22, 40, 41 (again showing contours of q, B, and m) for maximum admissible and minimum inadmissible numbers of connections, average rate lOkbit/s, average period 10 ms. Fig. 22 A shows case 21 ( ONOFF periodic, P_{ON}=0.01, 14999 connections); Fig. 22B shows case 22 (discrete periodic, 14999 connections); Fig. 22C shows case 40 (ONOFF periodic, P_{ON}=0.01, 15000 connections); Fig. 22D shows case 24 (Poisson, 15000 connections). In Fig. 22 there is no difference between the ONOFF periodic and discrete periodic models of approximately the same type of connections. This is because the burst size 100 bit is much smaller than the buffer size 3 Mbit.
Fig. 23A  Fig. 23D illustrate traffic cases 23, 24, 42, 43 (again showing contours of q, B, and m) for maximum admissible and minimum inadmissible numbers of connections, average rate lOkbit/s, average period 10 ms. Fig. 23 A shows case 23 (ONOFF Markov, P_{ON}=0.01, 14996 connections); Fig. 23B shows case 24 (Poisson, 14998 connections); Fig. 23C shows case 42 (ONOFF Markov, P_{ON}=0.01, 14997 connections); Fig. 24D shows case 43 (Poisson, 14999 connections).
Fig. 24A  Fig. 24D illustrate traffic cases 25, 26, 44, 45 (again showing contours of q, B, and m) for maximum admissible and minimum inadmissible numbers of connections, average rate lOkbit/s, average period 100 s. Fig. 24A shows case 25 ( ONOFF periodic, P_{ON}=0.01, 10409 connections); Fig. 24B shows case 26 (discrete periodic, 84 connections); Fig. 24C shows case 44 (ONOFF periodic, P_{ON} ^{=}0.01 , 10410 connections); Fig. 24D shows case 45 (discrete period, 85 connections). In Fig. 24 there is a great difference between the ONOFF periodic and discrete periodic models of approximately the same type of connections. This is because the burst size 1Mbit is of the same order of magnitude as the buffer size 3 Mbit. Fig. 25A  Fig. 25D illustrate traffic cases 27, 28, 46, 47 (again for contours of q, B, and m) for maximum admissible and minimum inadmissible numbers of connections, average rate lOkbit/s, average period 100 s. Fig. 25 A shows case 27 (ON OFF Markov, P_{ON}=0.01, 10342 connections); Fig. 25B shows case 28 (Poisson, 81 connections); Fig. 25C shows case 46 (ONOFF Markov, P_{ON}=0.01, 10343 connections); Fig. 25D shows case 47 (Poisson, 82 connections).
Fig. 26A  Fig. 26D illustrate traffic cases 30, 31, 49, 50 (again showing contours of q, B, and m) for maximum admissible and minimum inadmissible numbers of connections, average rate lOkbit/s, average period 10 ms. Fig. 26A shows case 30 (ONOFF periodic, P_{ON}=0.01, 14 connections); Fig. 26B shows case 31 (discrete periodic, 14 connections); Fig. 26C shows case 49 (ONOFF periodic, P_{ON}=0.01, 15 connections); Fig. 26D shows case 50 (discrete period, 15 connections).
Fig. 27A  Fig. 27B illustrate traffic cases 32, 33, 51, 52 (again for contours of q, B, and m) for maximum admissible and minimum inadmissible numbers of connections, average rate lOMbit/s, average period 10 ms. Fig. 27A shows case 32 (ONOFF Markov, P_{ON}=0.01, 9 connections); Fig. 27B shows case 33 (Poisson, 11 connections); Fig. 27C shows case 51 (ONOFF Markov, P_{ON}=0.01, 10 connections); Fig. 27D shows case 52 (Poisson, 12 connections
Fig. 28A  Fig. 28D illustrate traffic cases 34, 35, 53, 54 (showing contours of q, B, and m) for maximum admissible and minimum inadmissible numbers of connections, average rate lOMbit/s, average period 100 s. Fig. 28 A shows case 34 (ONOFF periodic, P_{ON}=0.01, 1 connection); Fig. 28B shows case 35 (discrete periodic, 0 connections); Fig. 28C shows case 53 (ONOFF periodic, P_{ON}=0.01, 2 connections); Fig. 28D shows case 54 (discrete period, 1 connection; same as case 16; no B <=3 Mbit and no m<=0 found)
Fig. 29A  Fig. 29D illustrate traffic cases 36, 37, 55, 56 (showing contours of q, B, and m) for maximum admissible and minimum inadmissible numbers of connections, average rate lOMbit/s, average period 100 s. Fig. 29A shows case 36 (ONOFF Markov, P_{ON}=0.01, 1 connection; same as case 17); Fig. 29B shows case 37 (Poisson, 0 connections); Fig. 29C shows case 55 (ONOFF Markov, P_{ON}=0.1, 2 connections); Fig. 29D shows case 56 (Poisson; same as case 18). Fig. 30A  Fig. 30B illustrate traffic cases 38, 57 (showing contours of q, B, and m) for maximum admissible and minimum inadmissible numbers of connections, average rate lOMbit/s, always ON or OFF, P_{ON}=0.1 Fig. 30A shows case 38, 1 connection; same as case 19); Fig. 30B shows case 57 (2 connections).
While the invention has been described in connection with what is presently considered to be the most practical and preferred embodiment, it is to be understood that the invention is not to be limited to the disclosed embodiment, but on the contrary, is intended to cover various modifications and equivalent arrangements included within the spirit and scope of the appended claims.
Claims
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US6993018B1 (en)  19990803  20060131  Telefonaktiebolaget Lm Ericsson (Publ)  Priority signaling for cell switching 
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