We intend to derive analytical expressions for modeling throughput and delay characteristics of a MAC protocol that mimics the IEEE 802.11e in every essential respect. We do so by first proposing a simplified model of the IEEE 802.11e MAC.

### 2.1 System model

We set out to analyze the 802.11e MAC protocol. We realize that an analysis of the exact scheme is cumbersome. We thus propose a hybrid-MAC model that resembles the 802.11e MAC in most essential respects. Our MAC model provides us with an abstraction of the essential features of 802.11e MAC, while avoiding the complex details of the latter. We believe that the insights obtained using our model are applicable to the 802.11e scenario. Our system model can be thought of as a hybrid MAC model which operates in both the contention and CFPs alternately, akin to a legacy 802.11 MAC protocol [4] with both its (a) distributed coordination function (DCF) and (b) point coordination function (PCF) modes enabled [4]. While DCF is based on the contention-based CSMA/CA mode of channel access, PCF is based on the polling mechanism. Limited QoS support in the legacy 802.11 standard is available through the use of the PCF. The DCF phase mimics the enhanced distributed channel access (EDCA) mechanism which is a contention-based channel access scheme while the PCF mimics the HCCA which is based on a polling mechanism. EDCA and HCCA are used to provide prioritized and parameterized QoS services, respectively, in 802.11e.

The network topology being modeled consists of a BSS of *N* LP and *M* HP traffic flows. We assume that each flow is generated by a node which we refer to as a STA (station), as done in the 802.11 standard. During the contention period (CP), each STA uses the basic access mechanism only, that is, no STA is assumed to be hidden from another STA and the RTS/CTS mechanism is not employed. During the contention-free period (CFP), the *M* HP traffic STAs are placed in a circular queue and are polled sequentially by the PCF. The PCF implements two periods of channel access in a duration of time referred to as the "superframe": (i) a CFP and (ii) a CP. Figure 2 depicts an 802.11e superframe. The proportion of time allocated to each period within a superframe is not defined by the standard. The point coordinator subsystem residing in an AP continues to poll STAs in its polling list until the CFP duration expires.

### 2.2 Modeling throughput

Our analytical model for overall system throughput is a dimensionless multivariable function *S* of *N*, *M*, *p*, and α,

S=S\left(N,M,p,\alpha \right)

(1)

where *p* is the probability of a successful frame transmission and α is a value between 0 and 1 that identifies the ratio of the time spent in the CFP to the total time spanned by a superframe which forms a repeating interval of contention and CFPs,

\alpha =\frac{\mathsf{\text{CFP}}}{\mathsf{\text{CFP}}+\mathsf{\text{CP}}}

(2)

As α tends toward 0, the BSS reverts to a contention-only-based environment where the point coordinator is not used to poll STAs. With a non-zero α, dimensionless throughput *S* becomes a weighted sum of time spent in the CP and the CFP,

S\left(N,M,p,\alpha \right)=\left(1-\alpha \right){S}_{\mathsf{\text{CP}}}+\alpha {S}_{\mathsf{\text{CFP}}}

(3)

We then apply the definitions of *S*_{CP} and *S*_{CFP} given in [12] for dimensionless throughput for each respective period,

{S}_{\mathsf{\text{CP}}}=\frac{{\mathit{\u016a}}_{\mathsf{\text{CP}}}}{{\mathit{\u012a}}_{\mathsf{\text{CP}}}+{\stackrel{\u0304}{B}}_{\mathsf{\text{CP}}}}

(4)

{S}_{\mathsf{\text{CFP}}}=\frac{{\mathit{\u016a}}_{\mathsf{\text{CFP}}}}{{\stackrel{\u0304}{B}}_{\mathsf{\text{CFP}}}}

(5)

In Equation 4, {\overline{U}}_{\mathsf{\text{CP}}} is the average duration of time the useful data are received by a STA during the CP, {\overline{I}}_{\mathsf{\text{CP}}} is the average duration of time the channel remains idle during the CP, and {\overline{B}}_{\mathsf{\text{CP}}} is the average duration of time the channel is busy transmitting data, the overhead bits incurred by the data, and is handling collisions [12]. Equation 4 is then a dimensionless quantity between 0 and 1 that represents throughput efficiency as the ratio of time the channel is used for sending useful data to total time. We can extend this concept by defining *S*_{CFP} in a similar way, with the exception that we exclude the idle term in the denominator since it is assumed that the channel is never idle during the CFP. The definitions of {\overline{U}}_{\mathsf{\text{CP}}}, {\overline{I}}_{\mathsf{\text{CP}}}, and {\overline{B}}_{\mathsf{\text{CP}}} are extended from [12], with the modification that the total STA count has been replaced by (*N* + *M*),

{\mathit{\u016a}}_{\mathsf{\text{CP}}}=\frac{\left(N+M\right)Tp}{\left(1-p\right)\left(1-{\left(1-p\right)}^{N+M}\right)}

(6)

{\mathit{\u012a}}_{CP}=\frac{\sigma}{1-{\left(1-p\right)}^{N+M}}

(7)

{\stackrel{\u0304}{B}}_{\mathsf{\text{CP}}}=\frac{{T}_{\mathsf{\text{s}}}}{{\left(1-p\right)}^{N+M}}

(8)

In Equation 6, *T* is the time spent in the CP transmitting *useful* data, that is, the ratio of the length in bits of packet payload *P* (excluding the number of header and trailer bits, *H*) to the data rate *R*. The other time parameter, *T*_{s}, in (8) is the time spent sensing the channel during a successful frame transmission. Substituting (6), (7), and (8) into (4), we obtain, as in [12],

{S}_{\mathsf{\text{CP}}}=\frac{\left(N+M\right)Tp{\left(1-p\right)}^{N+M-1}}{{T}_{\mathsf{\text{s}}}+\left(\sigma +{T}_{\mathsf{\text{s}}}\right){\left(1-p\right)}^{N+M}}

(9)

The expression for *T*_{s} is given by

{T}_{\mathsf{\text{s}}}=\mathsf{\text{DIFS}}+\frac{H+P}{R}+\mathsf{\text{SIFS}}+\frac{\mathsf{\text{ACK}}}{R}+2\tau

(10)

To derive Equation 10, we note that synchronized data exchange within the CFP are accomplished by polling STAs. The polling process is coordinated by the PCF implementation within an AP. When the CFP begins, the AP waits a brief duration of time known as a *short interframe space* (SIFS) which serves as a delay between beacon, data, acknowledgement, and end frames that are transmitted during the CFP. The value of SIFS varies by the particular 802.11 standard implemented by a transceiver. For 802.11a, b, and g, the values are 16, 10, and 10 μs, respectively. After waiting an initial SIFS, the AP commences with polling by transmitting a Data/CF-Poll frame to the first STA in a polling list. Data/CF-Poll frames serve a dual purpose by piggybacking data carried by the AP which, in an infrastructure mode network, is attached to a wired network via a wired Ethernet interface. The Data/CF-Poll frame polls the receiving STA while simultaneously carrying higher layer datagrams originating from another STA within a BSS or a device external to a BSS via a wired LAN. The collision avoidance (CA) mechanism of CSMA/CA cannot guarantee collisions will not occur. A collision can occur, for example, if two STAs compute exactly the same backoff time after detecting a channel idle for DCF interframe space duration (DIFS) and then transmit a MPDU when the backoff timer matures. To determine if a transmission resulted in a collision, each data frame (MPDU) must be acknowledged through the transmission of an ACK frame sent by the STA receiving a data frame. If a sending STA does not receive a corresponding ACK after waiting a SIFS period, the sending STA concludes a collision occurred and will repeat the transmission. DIFS values for 802.11a, b, and g are 34, 50, and either 28 or 50 μs, depending on slot time, respectively. In IEEE 802.11g, the slot time can be either 9 μs if no legacy 802.11b STAs are present in the BSS, or 20 μs if the BSS has a mix of 802.11b and 802.11g STAs. DIFS is a function of SIFS and is computed according to

\mathsf{\text{DIFS}}=\mathsf{\text{SIFS}}+2\sigma

(11)

where σ is the slot time defined to be twice the maximum propagation time τ. The slot time is therefore an amount of time a STA requires to determine if another STA has accessed the channel at the start of the previous slot. Slot time values for 802.11a and b are 9 and 20 μs, respectively, for a PHY that uses a direct sequence spread spectrum (DSSS) modulation technique and 50 μs for a PHY that uses a frequency hopping spread spectrum (FHSS) transmission method. Acknowledgement frames may also piggyback data originating from a receiving STA and intended for another STA in the BSS or an external device. If the point coordinator fails to receive a response from a polled STA within a PCF interframe space (PIFS) period of time, the PCF will move on and poll the next STA in its polling list. PIFS is also function of SIFS and is computed according to

\mathsf{\text{PIFS}}=\mathsf{\text{SIFS}}+\sigma

(12)

and thus the values for 802.11a, b, and g are 25, 30, and either 19 or 30 μs, respectively. The PIFS duration also serves as a gap between the CP and CFP. From (11) and (12) we have the following inequality

\mathsf{\text{SIFS}}<\mathsf{\text{PIFS}}<\mathsf{\text{DIFS}}

(13)

which prevents the PCF from transmitting a poll frame in between a Data/CF-Poll and Data/CF-ACK transaction.

Given the definitions of SIFS and DIFS, Equation 10 can be understood as the sum of times required to conduct a successful packet transmission in the CP: the STA must first wait a DIFS amount of time to detecting a channel idle before proceeding to transmit, then an (*H* + *P*)/*R* amount of time to for an interface to transmit a packet consisting of *H* header and trailer bits and *P* payload bits at a data rate *R*, then a τ amount of time for propagation of the data packet, then a SIFS amount of time before the receiving STA's interface can transmit an acknowledgement frame, then (ACK/*R*) time to transmit the acknowledgement frame, and finally another τ amount of time for propagation of the acknowledgement.

Our derivation of *S*_{CFP} proceeds in a similar way to that of *S*_{CP}. Let *q* represent the probability a STA has a non-null data frame to transmit during the CFP. {\overline{U}}_{\mathsf{\text{CFP}}} is the average time spent during the CFP to transmit *useful* data. By useful data we mean data bits and not bits belonging to beacon, pure ACK, and CF-End frames. If we denote *P*_{CFP} as the number of data bits transmitted during the CFP, then

{\mathit{\u016a}}_{\mathsf{\text{CFP}}}=\frac{{P}_{\mathsf{\text{CFP}}}}{R}

(14)

where *R* is the fixed transceiver data rate.

To derive an expression for the mean time the channel is busy in the CFP during a successful polling transaction, denoted {\overline{B}}_{\mathsf{\text{CFP}}}, we need to account for all the individual frame transmissions namely, CF_{Beacon}, CF_{Poll}, CF_{ACK}, and CF_{Null} which represent the lengths of the beacon, Data/CF-Poll, Data/CF-ACK, and CF-NULL frames, respectively. CF-Null frames are transmitted by a polled STA if the STA does not have any pending data to send, τ is the propagation delay of the wireless LAN, and *H* is the length of the header and frame check sequence (FCS) of an 802.11 frame. The first term in Equation 15 is the time required for the hybrid coordinator (HC) operating in an access point to transmit a beacon frame and for the beacon to propagate. The second term in (15) is the time required to poll all the LP and HP stations being coordinated by the HC during the CFP. The third term is the probability all the stations have a non-null data frame waiting to transmit upon being polled. The summation in parenthesis is the time required for the corresponding station to acknowledge the poll by returning a combined Data/CF-ACK frame. The fourth term then accounts for the time required for all the stations that do *not* have data to send and will transmit a CF-NULL frame back to the HC upon being polled.

\begin{array}{cc}\hfill {\stackrel{\u0304}{B}}_{\mathsf{\text{CFP}}}& =\left(\mathsf{\text{PIFS}}+\frac{\mathsf{\text{C}}{\mathsf{\text{F}}}_{\mathsf{\text{Beacon}}}}{R}+\tau \right)+\hfill \\ \left(N+M\right)\left(\mathsf{\text{SIFS}}+\frac{H+P+\mathsf{\text{C}}{\mathsf{\text{F}}}_{\mathsf{\text{Poll}}}}{R}+\tau \right)+\hfill \\ \left(N+M\right){q}^{\left(N+M\right)}\left(\mathsf{\text{SIFS}}+\frac{H+P+\mathsf{\text{C}}{\mathsf{\text{F}}}_{\mathsf{\text{Data/ACK}}}}{R}+\tau \right)+\hfill \\ \left(N+M\right){\left(1-q\right)}^{\left(N+M\right)}\left(\mathsf{\text{SIFS}}+\frac{H+P+\mathsf{\text{C}}{\mathsf{\text{F}}}_{\mathsf{\text{Null}}}}{R}+\tau \right)+\hfill \\ \left(\mathsf{\text{SIFS}}+\frac{\mathsf{\text{C}}{\mathsf{\text{F}}}_{\mathsf{\text{End}}}}{R}+\tau \right)\hfill \end{array}

(15)

### 2.3 Modeling delay

Our analytical model for overall system delay is a dimensionless multivariable function *D* of *N*, *M*, *p*, and α,

D=D\left(N,M,p,\alpha \right)

(16)

Observe that

0<\frac{{D}_{\mathsf{\text{ideal}}}}{{D}_{\mathsf{\text{actual}}}}\le 1

(17)

where *D*_{ideal} is the theoretical minimum delay a STA can experience in a superframe while *D*_{actual} is the true delay experienced. If we define *D* such that

D=\left(1-\frac{{D}_{\mathsf{\text{ideal}}}}{{D}_{\mathsf{\text{actual}}}}\right)

(18)

Then *D* → 0 as the actual delay approaches the ideal and *D* → 1 as actual delay diverges from the ideal. We first consider delay incurred by the DCF. Ideal delay in the CP can be expressed as the sum of ideal HOL delay and ideal queuing delay,

{D}_{\mathsf{\text{ideal}}}={D}_{\mathsf{\text{ideal}}}^{\mathsf{\text{HOL}}}+{D}_{\mathsf{\text{ideal}}}^{\mathsf{\text{Queuing}}}

(19)

where {D}_{\mathsf{\text{ideal}}}^{\mathsf{\text{HOL}}} represents the minimum time required in the CP to transmit an 802.11 frame successfully, upon the first attempt, and is equal to *T*_{s}. Ideal queuing delay is given by the Pollaczek-Khinchine formula [12]

{D}_{\mathsf{\text{ideal}}}^{\mathsf{\text{Queuing}}}=\frac{\rho}{2\mu \left(1-\rho \right)}\left(1+c{v}^{2}\right)

(20)

that describes the mean time a frame waits in queue to be serviced by the MAC, where the queue is modeled as a M/G/1 queue (a single server with frame arrivals having a Poisson distribution and service time having a general distribution). Total actual delay *D*_{actual} is modeled as the sum of (20) and an expression for the expected value of HOL delay which takes into account backoff delay.

In Equation 21, β is the average physical time between two decrements of the backoff counter, CW_{min} is the minimum contention window size, {P}_{s}={\left(1-p\right)}^{M+N-1} is the probability a STA's frame transmission is successful, and *r*_{max} is the maximum number of retransmissions permitted. In our simulation, CW_{min} is set to 2^{4} and CW_{max} is set to 2^{10} which are the values used by a PHY that employs a FHSS method of transmitting radio signals. Considering now the PCF, each STA has an opportunity to transmit when polled while the CFP is in progress. If the maximum predetermined duration of the CFP in a given superframe expires before every STA has been polled, STAs that were not given an opportunity are more likely to be polled in the following CFP as the PC uses a circular queue to schedule station polling.

\begin{array}{cc}\hfill E\left[{D}_{\mathsf{\text{actual}}}^{\mathsf{\text{HOL}}}\right]& ={T}_{\mathsf{\text{s}}}+\beta \left[\frac{\mathsf{\text{C}}{\mathsf{\text{W}}}_{min}}{2\left(1-{\left(1-{P}_{s}\right)}^{{r}_{max}+1}\right)}\right]\hfill \\ \left[\frac{{P}_{s}\left(1-{\left(2\left(1-{P}_{s}\right)\right)}^{{r}_{max}+1}\right)}{1-2\left(1-{P}_{s}\right)}-1-{\left(1-{P}_{s}\right)}^{{r}_{max}+1}\right]+\hfill \\ {T}_{\mathsf{\text{s}}}\left[\frac{1-{P}_{s}}{{P}_{s}}\right]\left[\frac{{\left(1-{P}_{s}\right)}^{{r}_{max}}\left(-{P}_{s}{r}_{max}-1\right)+1}{1-{\left(1-{P}_{s}\right)}^{{r}_{max}+1}}\right]\hfill \end{array}

(21)

Also, *r*_{max} is defined as

{r}_{\mathrm{max}}={\text{log}}_{2}\left({\text{CW}}_{\mathrm{max}}/{\text{CW}}_{\mathrm{min}}\right)

(22)

since the number of different contention window sizes will be the exponent of the ratio of CW_{max} to CW_{min}. Equation (22) therefore gives the maximum number of retransmission attempts that will be made, if the initial transmission should result in a collision. For a FHSS based PHY, *r*_{max} is 6.

{D}_{\mathsf{\text{ideal}}}^{\mathsf{\text{HOL}}} in (21) is without any backoff delay,

{D}_{\mathsf{\text{ideal}}}^{\mathsf{\text{HOL}}}={T}_{\mathsf{\text{s}}}

(23)

Let *ψ* be a random variable and *E*[ψ] represent the expected value (a number in the range [0, 2312]) of the size of the body of data within an 802.11 frame transmitted by a polled station during the CFP, then

\text{\Psi}=34+E\left[\psi \right]

(24)

since 34 equals the maximum number of bits that comprise an 802.11 MAC header with the cyclic redundancy check (CRC) (A.K.A FCS) field included (see Figure 3).

Assuming the length of data in frames transmitted during the CFP follows a discrete uniform distribution (i.e., all frame lengths within the range [0,2312] are equally likely), \overline{\text{\Psi}}=E[\text{\Psi}]=34+\left(0+2312\right)/2=1190 bits and the mean total time for one CFP is given by {\overline{T}}_{\mathsf{\text{CFP}}},

\begin{array}{c}{\overline{T}}_{\text{CFP}}=\text{PIFS}+\frac{{\text{CF}}_{\text{Beacon}}}{R}+\left(N+M\right)\frac{\left({\overline{\text{\Psi}}}_{\text{PC}}+{\overline{\text{\Psi}}}_{\text{STA}}\right)}{R}\\ +\left[2\left(N+M\right)+1\right]\text{SIFS}+\frac{{\text{CF}}_{\text{End}}}{R}+2\left[N+M+1\right]\tau \end{array}

(25)

In Equation 25, we account for polling frames that may either be CF-Poll with no data (subtype 6 or 0110) or CF-Poll + Data (subtype 2 or 0010) as \left(N+M\right){\stackrel{\u0304}{\text{\Psi}}}_{\mathsf{\text{PC}}} represents the mean length of polling frame bits transmitted by the point coordinator during the CFP. Similarly, we account for acknowledgement frames that may be CF-ACK with no data (subtype 5 or 0101) or CF-ACK + Data (subtype 1 or 0001) as \left(N+M\right){\stackrel{\u0304}{\text{\Psi}}}_{\mathsf{\text{STA}}} represents the mean length of acknowledgement frame bits transmitted by all the stations during the CFP. The remaining terms in (25) follow from (15) and account for interframe delays, management and control frames, and propagation times.

Let *D*_{CFP} represent the average time a frame must wait at the HOL once the CFP begins. The first polled station must wait

\mathsf{\text{PIFS}}+\frac{\mathsf{\text{C}}{\mathsf{\text{F}}}_{\mathsf{\text{Beacon}}}}{R}+2\left(\mathsf{\text{SIFS}}+\tau \right)+\frac{{\text{\Psi}}_{\mathsf{\text{PC}}}}{R}

(26)

time duration before transmitting a frame. The second station must wait the time given in (26) plus

2\left(\mathsf{\text{SIFS}}+\tau \right)+\frac{{\text{\Psi}}_{\mathsf{\text{STA}}}+{\text{\Psi}}_{\mathsf{\text{PC}}}}{R}

(27)

amount of time before transmitting a frame. Thus, from (26) and (27), the average time a station must wait before transmitting a frame is

\begin{array}{cc}\hfill {\stackrel{\u0304}{D}}_{\mathsf{\text{CFP}}}& =\mathsf{\text{PIFS}}+\left(N+M\right)\left(\mathsf{\text{SIFS}}+\tau \right)+\frac{\mathsf{\text{C}}{\mathsf{\text{F}}}_{\mathsf{\text{Beacon}}}}{R}\hfill \\ +\frac{1}{R}\left[\left(\frac{N+M}{2}\right){\text{\Psi}}_{\mathsf{\text{PC}}}+\left(\frac{N+M}{2}-1\right){\text{\Psi}}_{\mathsf{\text{STA}}}\right]\hfill \end{array}

(28)

From (19), (20), (23), and (28) we now have

{\stackrel{\u0304}{D}}_{\mathsf{\text{ideal}}}={T}_{\mathsf{\text{s}}}+\frac{\rho}{2\mu \left(1-\rho \right)}\left(1+c{v}^{2}\right)+{\stackrel{\u0304}{D}}_{\mathsf{\text{CFP}}}

(29)

Accounting for backoff delay, the actual delay is modified to give *D*_{actual} which is shown in (25).

\begin{array}{cc}\hfill {\stackrel{\u0304}{D}}_{\mathsf{\text{actual}}}& ={T}_{\mathsf{\text{s}}}+\beta \left[\frac{\mathsf{\text{C}}{\mathsf{\text{W}}}_{min}}{2\left(1-{\left(1-{P}_{s}\right)}^{{r}_{max}+1}\right)}\right]\hfill \\ \left[\frac{{P}_{s}\left(1-{\left(2\left(1-{P}_{s}\right)\right)}^{{r}_{max}+1}\right)}{1-2\left(1-{P}_{s}\right)}-1-{\left(1-{P}_{s}\right)}^{{r}_{max}+1}\right]\hfill \\ +{T}_{s}\left[\frac{1-{P}_{s}}{{P}_{s}}\right]\left[\frac{{\left(1-{P}_{s}\right)}^{{r}_{max}}\left(-{P}_{s}{r}_{max}-1\right)+1}{1-{\left(1-{P}_{s}\right)}^{{r}_{max}+1}}\right]\hfill \\ +\frac{\rho}{2\mu \left(1-\rho \right)}\left(1+c{v}^{2}\right)\hfill \\ +\mathsf{\text{PIFS}}+\left(N+M\right)\left(\mathsf{\text{SIFS}}+\tau \right)+\frac{\mathsf{\text{C}}{\mathsf{\text{F}}}_{\mathsf{\text{Beacon}}}}{R}\hfill \\ +\frac{1}{R}\left[\begin{array}{c}\mathsf{\text{C}}{\mathsf{\text{F}}}_{\mathsf{\text{Beacon}}}+\left(\frac{N+M}{2}\right){\text{\Psi}}_{\mathsf{\text{PC}}}\\ +\left(\frac{N+M}{2}-1\right){\text{\Psi}}_{\mathsf{\text{STA}}}\end{array}\right]\hfill \end{array}

(30)

### 2.4 Analysis of the hybrid-protocol simulation results

We evaluated the accuracy of our analytical expressions for dimensionless throughput and normalized delay by developing a MATLAB simulation based on our derivations. Figures 3 and 4 represent the dimensionless throughput and normalized delay values as the number of HP STAs in the BSS increases with varying superframe period duration α. We see that the value of α has a significant effect on system performance with respect to throughput and delay. Similarly, the collision probability impacts throughput and delay. Figures 3 and 4 show a surface plot that quantifies the relationship between collision probability, number of HP users, and the effect these parameters have on system delay and throughput, respectively. When the system operates in equal duration of CP and CFP (i.e., α = 0.5), the throughput decreases with an increase in the number of HP users, gradually approaching an asymptote. This can be explained by the fact that an increasing number of HP users create higher contention in the CP phase leading to longer backoff time and thereby a drop in throughput and an increase in delay, as seen in Figure 4. Interestingly, the delay value also approaches an asymptote as the number of HP users in the BSS increase (when α = 0.5). In Figure 3, we see that, as the number of HP stations increases, a saturation condition at normalized delay *D* = 1 is attained with lower values of collision probability *p*.

With respect to Figures 4 and 5, collision probability *p* is defined as the probability a given frame transmission attempt is unsuccessful due to a collision occurring in the CP. Looking at Figure 3, one can see that for a small number of HP stations, the directional derivative *dD/dp* is much less than it is for a large number of HP stations. Because the rate of change in delay increases faster with respect to station count as collision probability increases, a saturation condition will arise sooner in a BSS with many high priority traffic stations if stations begin to experience a greater number of collisions in the contention period. Similarly, in Figure 4, we see how small changes in collision probability can greatly affect throughput as the HP station count increases. We also see the appearance of an optimal throughput contour along the maxima of the surface *S*.