Data-Unit-Size Distribution Model with Retransmitted Packet Size Preservation Property and Its Application to Goodput Analysis for Stop-and-Wait Protocol: Case of Independent Packet Losses
aa r X i v : . [ c s . PF ] O c t Data-Unit-Size Distribution Model with Retransmitted Packet Size PreservationProperty and Its Application to Goodput Analysis for Stop-and-Wait Protocol:Case of Independent Packet Losses
Takashi Ikegawa
Abstract
This paper proposes a data-unit-size distribution model to represent the retransmittedpacket size preservation (RPSP) property in a scenario where independently lost packets areretransmitted by a stop-and-wait protocol. RPSP means that retransmitted packets with thesame sequence number are equal in size to the packet of the original transmission, which isidentical to the packet generated from a message through the segmentation function, namely,generated packet. Furthermore, we derive goodput formula using an approach to derive thedata-unit-size distribution. We investigate the effect of RPSP on frame size distributions andgoodput in a simple case when no collision happens over the bit-error prone wireless networkequipped with IEEE 802.11 Distributed Coordination Function, which is a typical example ofthe stop-and-wait protocol. Numerical results show that the effect gets stronger as bit errorrate increases and the maximum size of the generated packets is larger than the mean sizefor large enough packet retry limits because longer packets will be repeatedly corrupted andretransmitted more times as a result of RPSP.
Index Terms
Data unit size, retransmitted packet size preservation property, message segmentation,goodput, independent packet loss, IEEE 802.11 Distributed Coordination Function.
I. I
NTRODUCTION
Transfers of data units over communication networks suffer frequently from failure dueto various reasons including bit errors, congestion and collision. To provide an error-freetransmission service of messages, i.e., data units generated by reliable applications, a senderrequires to implement one or more communication protocols that include error recovery function.The error recovery function allows the sender to retransmit lost packets. For example, distributedcoordination function (DCF) for IEEE 802.11 wireless local area networks specifies a stop-and-wait protocol (SWP) to realize the error-recovery function in a simple manner [1].The packets, i.e., SWP-layer data units, that have been corrupted or lost within the networkswill be transmitted by the error-recovery function. In general, such retransmitted packets withthe same sequence number ( seqNum ) are equal in size to the packet in the original transmission.We call this property retransmitted packet size preservation: RPSP.The packet retransmission probability will depend on the size of frames, which are data unitsthat contain the packet and are transferred over physical links. Typical situations include the casewhen frames are lost due to bit errors because the frame corruption probability is approximatelyproportional to the frame size.In papers [2]–[4], the effect of RPSP on the mean frame size was discussed for bit errorprone networks. These papers showed that the mean frame size with RPSP is larger than thatwithout RPSP as bit error rate increase if the packet size distribution has dispersion. The reasonfor this is that longer frames will be repeated corrupted more time due to RPSP.The frame sizes affect several quality of service (QoS) parameters (e.g., goodput) for applica-tions. Consequently, the effect of RPSP on QoS parameters will appear in some cases. However,
T. Ikegawa is with Waseda Research Institute for Science and Engineering, Waseda University and Graduate Schoolof Mathematical Sciences, the University of Tokyo, Japan (e-mail: [email protected] or [email protected]).
MT RCVMessages
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Fig. 1C
OMMUNICATION NETWORK MODEL . in previous work on QoS parameter analysis over links with bit errors, such as studies for IEEE802.11 DCF goodput analysis including [5]–[7], the effect of RPSP was ignored. For example,frame sizes are assumed to be constant although actual frames size distribution has dispersion(e.g. [8], [9]). The purpose of this paper is to propose a data-unit-size distribution model withRPSP to represent among the sizes of respective data units (i.e., messages, generated packets,transferred packets and frames) and to derive the goodput formula using an approach to derivethe data-unit-size distribution.The rest of the paper is organized as follows. In the next section, we describe the communi-cation network model underlying our study. Section III derives the forms of size distributionsof generated packets, transferred packets and frames. Section IV derives the form of goodputand applies the result to an IEEE 802.11 DCF wireless network. Section V investigates theeffect of RPSP on the frame size distribution and goodput for actual message-size distributionsFinally, Section VI summarizes this paper and mentions future work. II. C
OMMUNICATION NETWORK MODEL
In this section, we first explain the three-layered communication network model under con-sideration. Next, the model of data units introduced in this paper at the respective layer isdescribed. In final, we explain some assumptions for analytical tractability.
A. Layer model
To characterize the nature of RPSP and message segmentation, we consider a communicationnetwork of which conceptual representation is shown in Fig. 1. Each station (a sender and areceiver) has three layers. The middle layer is referred to as an SWP layer. It implementsmessage segmentation-reassembly and error-recovery functions. The error-recovery function isassumed to be implemented in a stop-and-wait scheme. The layer above the SWP layer, namelythe higher layer, contains a traffic source and sink. The traffic source generates the data units.On the other hand, the traffic sink terminates the corresponding data units. The layer below theSWP layer, namely the lower layer, contains an entity that can transfer data units over physicallinks at a sender.2 . Data-unit model
We define data units exchanged between peer entities at the respective layer as follows:
Message: a data unit generated by a traffic source with a given size distribution ofwhich function is denoted by F ( m ) ( · ) . Packet: a data unit created from a message through segmentation function by addinga header and/or trailer, i.e., control information, to the (divided) message. We assumethat size of SWP-layer’s control information is constant and equal to ℓ (R) h . Whenevera packet is created, a seqNum ( ≥ ) is assigned. To model the RPSP explicitly, thepackets are categorized into the following two kinds: Generated packet: a packet that is generated from a message by a sender’sSWP layer at the original transmission. The message segmentation functionimplemented in the sender’s SWP layer enables a single message to be dividedinto several generated packets if the message size is larger than the payloadsize ℓ d ( > . The receiver’s SWP layer performs a message reassemblyfunction, thus reassembling the segmented generated packets before deliveringthem to the higher layer. Transferred packet: a packet that is encapsulated into the frame. Due toRPSP, all the sizes of transferred packets with the same seqNum are equalto that of the generated packet.
Frame: a data unit that is made by encapsulating a transferred packet into a frameand by adding control information to the transferred packet, and will be transferredover physical links. The size of lower-layer’s control information is assumed to becosntant and equal to ℓ (L) h . C. Assumptions
For analytical tractability, we make the following assumptions. A1 : Message sizes are mutually independent and identically distributed according to acommon message-size distribution function F ( m ) ( · ) . The distribution F ( m ) ( · ) has afinite mean value ℓ ( m ) , which is referred to as the mean message size. A2 : Frames are independently lost with probability g ( x ) , ≤ g ( x ) < , where x is the size of information field in the frame, equivalently, the size of atransferred packet. A3 : The sender operates under a heavy traffic assumption, meaning that the sender’s SWPlayer always has a generated packet available to be sent. Example 1
Case of independent bit error prone links.
Typical situations satisfying assumption A2 include the cases where frames are lost due to bit errors that occur independently. Letting p e be bi-error rate, g ( x ) is given by g ( x ) = 1 − (1 − p e ) x + ℓ (L) h , (1)where x is the size of transferred packets. III. A
NALYSIS OF SIZE DISTRIBUTIONS FOR GENERATED PACKETS , TRANSFERRED PACKETS AND FRAMES
In this section, we derive the forms of a size distributions of generated packets, transferredpackets and frames under assumptions mentioned in the preceding section. 3 . Form of generated packet size distribution
Let random variable L ( p ) be a size of generated packets. Denoting F ( p ) ( · ) be a generatedpacket size distribution, that is F ( p ) ( x ) △ = Pr . ( L ( p ) ≤ x ) , from the argument [10], we have F ( p ) ( x ) = (cid:16) − π ( E ) (cid:17) ( x − ℓ d − ℓ (R) h ) + π ( E ) F ( E ) ( x ) , (2)where π ( E ) is an occurrence probability of edge packets and F ( E ) ( · ) is a distribution of edge-packet sizes. The edge packet is defined as the final segmented generated-packet, if a messageis segmented. It is identical with the original message if not segmented.The forms of π ( E ) and F ( E ) ( · ) are given by π ( E ) = 1 ∞ X s =0 Z ∞ sℓ d dF ( m ) ( x ) = 1 ∞ X s =0 (cid:16) − F ( m ) ( sℓ d ) (cid:17) , (3)and F ( E ) ( x ) = , ≤ x < ℓ (R) h , ∞ X s =0 n F ( m ) ( x + s ℓ d − ℓ (R) h ) − F ( m ) ( s ℓ d ) o , ℓ (R) h ≤ x ≤ ℓ d + ℓ (R) h , , x > ℓ d + ℓ (R) h . (4) Example 2
Case of discrete message-size distribution.
Consider the case where the message-size distribution function F ( m ) ( · ) is given by F ( m ) ( x ) = n d X i =1 ω ( m ) i ( x − ℓ ( m ) i ) , (5)where n d ≥ , w ( m ) i > , ℓ ( m ) i > for i = 1 , , · · · , n d , and P n d i =1 w ( m ) i = 1 .The form of π ( E ) is given by { P n d i =1 w ( m ) i k i } − with k i = ⌈ ℓ ( m ) i /ℓ d ⌉ . This can be intuitivelyshown from the fact that 1) k i generated packets are created from one message of size ℓ ( m ) i ,and 2) they consist of k i − generated packets of size ℓ d (called body packets [10]) and oneedge packet. The generated-packet-size distribution can be written as F ( p ) ( x ) = (cid:16) − π ( E ) (cid:17) ( x − ℓ d − ℓ (R) h ) + π ( E ) n d X i =1 w ( m ) i ( x − ℓ ( m ) i + ( k i − ℓ d − ℓ (R) h ) . (6)The form of (6) can be rewritten as F ( p ) ( x ) △ = n d X i =0 w ( p ) i ( x − ℓ ( p ) i ) , (7)where ( w ( p )0 = 1 − π ( E ) ,l ( p )0 = ℓ d + ℓ (R) h , (8) ( w ( p ) i = π ( E ) w ( m ) i ,l ( p ) i = ℓ ( m ) i − ( k i − ℓ d + ℓ (R) h , i = 1 , , · · · , n d . (9)Letting ℓ ( p ) be the mean packet size, we have ℓ ( p ) △ = Z ∞ x dF ( p ) ( x ) = π ( E ) ℓ ( m ) + ℓ (R) h . (10)4 . Form of transferred packet size distribution Let F ( q ) ( · ) be a transferred packet size distribution. Denoting the number of retransmissionsof the transferred packet with the same seqNum of the generated packet which size is equalto L ( p ) by R , we can prove the following proposition. Proposition 1
The transferred packet size distribution F ( q ) ( · ) is given by F ( q ) ( y ) = , ≤ y < ℓ (R) h , Z x = yx = ℓ (R) h E h R + 1 | L ( p ) = x i dF ( p ) ( x ) E [ R + 1] , ℓ (R) h ≤ y ≤ ℓ d + ℓ (R) h , , y > ℓ d + ℓ (R) h . (11) Proof:
See Appendix I.From assumption A2 , the form of E [ R + 1 | L ( p ) = x ] for ℓ (R) h ≤ x ≤ ℓ d + ℓ (R) h is given by E h R + 1 | L ( p ) = x i = (1 − g ( x )) n RL X r =0 ( r + 1) { g ( x ) } r + { g ( x ) } n RL +1 ( n RL + 1)= 1 − { g ( x ) } n RL +1 − g ( x ) △ = h ( x, n RL ) . (12)where n RL ( ≥ is the maximum number of retransmission attempts of the transferred packetwith the same seqNum , referred to as retry limit. Remark 1
Substitution of (2) into (11) yields F ( q ) ( y ) for ℓ (R) h ≤ y ≤ ℓ d + ℓ (R) h given by F ( q ) ( y ) = (cid:0) − π ( E ) (cid:1) h ( ℓ d + ℓ (R) h , n RL ) ( x − ℓ d − ℓ (R) h ) + π ( E ) Z x = yx = ℓ (R) h h ( x, n RL ) dF ( E ) ( x ) E [ R + 1] , (13)where E [ R + 1] is given by E [ R + 1] = (cid:16) − π ( E ) (cid:17) h ( ℓ d + ℓ (R) h , n RL ) + π ( E ) Z ℓ d + ℓ (R) h ℓ (R) h h ( x, n RL ) dF ( E ) ( x ) . (14) Example 3
RPSP effect when no frame is lost.
Consider the case where no frame is lost. Inthis case, the number of retransmissions is equal to zero, i.e., R = 0 . From (11), F ( q ) ( x ) isidentified with F ( p ) ( x ) , implying that no effect of RPSP appears. Example 4
RPSP effect when generated packets are constant in size.
Let us consider the casewhere generated packets have a common size ℓ c (= ℓ ( p ) ) , that is F ( p ) ( x ) = ( x − ℓ c ) . (15)Thypical situations include when message sizes follow the discrete distribution function givenby (5) with n d = 1 and ℓ ( m )1 (= ℓ c − ℓ (R) h ) ≤ ℓ d , resulting in π ( E ) = 0 . Note that F ( p ) ( x ) canbe approximated by ( x − ℓ d ) if ℓ ( m ) is large enough compared with ℓ d from [10, Remark 3].With (11) and (15), F ( q ) ( x ) is identified with F ( p ) ( x ) = ( x − ℓ c ) , which indicates that noeffect of RPSP appears. 5 . Form of frame size distribution Denote the frame size distribution by F ( f ) ( · ) . Since a frame contains a transferred packetand the size of control information added the transferred packet is ℓ (L) h , F ( f ) ( x ) is simply givenby F ( q ) ( x − ℓ (L) h ) . IV. G
OODPUT A NALYSIS
In this section, first, we derive the form of goodput in a simple scenario. Next, we apply theresult to an IEEE 802.11 DCF wireless network.
A. Form of goodput
Let G be goodput of a single SWP connection, which is defined as the mean number of bitsby a receiver’s higher layer entity across the higher layer interface per unit time. We denotethe interdeparture time of the transferred packet by T ( cycle ) . In addition, we denote the eventmeaning that the transferred packet is successfully transmitted by “ delivery ”. Then we can provethe following proposition. Proposition 2
The form of goodput G is given by G = Z ℓ d + ℓ (R) h x = ℓ (R) h Pr . ( delivery | L ( p ) = x ) ( x − ℓ (R) h ) dF ( p ) ( x ) Z ℓ d + ℓ (R) h x = ℓ (R) h E h R + 1 | L ( p ) = x i E h T ( cycle ) | L ( p ) = x i dF ( p ) ( x ) . (16) Proof:
See Appendix II.Note that assumption A2 yields the form of Pr . ( delivery | L ( p ) = x ) given by Pr . ( delivery | L ( p ) = x ) = 1 − { g ( x ) } n RL +1 . (17) B. Application of goodput analysis to IEEE 802.11 DCF
We consider a simple scenario where just one sender and one receiver exist in a wirelessnetwork equipped with IEEE 802.11 DCF, which is an SWP protocol. Since no collision occurs,from the argument described in [6], the form of E [ T ( cycle ) | L ( p ) = x ] in (16) can be simplywritten as E h T ( cycle ) | L ( p ) = x i = (1 − g ( x )) σ − { g ( x ) } n RL +1 n RL X r =0 b r { g ( x ) } r + (1 − g ( x )) t suc ( x ) + g ( x ) t bit ( x ) , (18)where σ : DCF backoff slot size b r : mean value of the backoff counter of the r th backoff stage, i.e., the r th retransmissionattempt of the transferred packet t suc ( x ) and t bit ( x ) :mean interdeparture times of the transferred packet of size of x when a transmissionis successful and fails due to bit errors, respectively.The value of E [ b r ] is equal to CW r / because the backoff time at each transmission isuniformly chosen in the range [0 , CW r ] where CW r is min { r ( CW min + 1) − , CW max } for r = 1 , , · · · , n RL and CW is CW min . Assuming that propagation delay is negligible, we have t suc ( x ) = x + ℓ ACK µ d + 2 ℓ (L) h µ b + t SIFS + t DIFS , (19)6 bit ( x ) = xµ d + ℓ (L) h µ b + t EIFS , (20)where µ d is data-transmission rate, µ b is basic-link rate, and ℓ ACK is ACK-packet size. Here, t SIFS , t DIFS and t EIFS are Short Inter Frame Space (IFS), DCF IFS and Extended IFS, respec-tively. The derivation of (18) can be found in Appendix III.
V. N
UMERICAL RESULTS AND DISCUSSIONS
In this section, we examine the effect of RPSP on frame-size distributions and goodput byutilizing the results in Sections III and IV. We consider a scenario in which Web objects aretransferred over the IEEE 802.11 DCF network where bit errors occur independently. In thefollowing, we used the parameter values listed in Table I.
TABLE IP
ARAMETER VALUES USED TO OBTAIN NUMERICAL RESULTS
Parameter ValueBasic-link rate µ b MbpsData-transmission rate µ d MbpsSWP layer information field size ℓ ( R ) h bytesLower layer information field size ℓ ( L ) h bytesSlot time σ µ secShort IFS t SIFS µ secDCF IFS t DIFS µ secExtended IFS t EIFS µ secACK-packet size ℓ ACK bytesMinimum contention window size CW min Maximum contention window size CW max
Two kinds of Web pages are considered: static and dynamic Web pages. We shall use thefollowing Web object size distributions from traffic measurements [8], [11]. • Static Web objects:
The sizes of the static Web objects is assumed to follow a lognormaldistribution given by F ( m ) ( x ) = Z y = xy =0 √ πσye − (log y − µ ) σ dy, x > , , x ≤ . (21)The distribution parameters µ and σ are assumed to be . and . , respectively, on thebasis of the measured mean message size ℓ ( m ) = 4827 bytes and the measured standarddeviation σ ( m ) = 41 , bytes. Note that this lognormal distribution can represent a long-tailed property. • Dynamic Web objects:
The sizes of the dynamic Web objects are assumed to follow aWeibull distribution: F ( m ) ( x ) = ( − e − ( λ x ) ν , x > , , x ≤ . (22)The scale parameter λ and the shape parameter ν are assumed to be . × − and . ,respectively, which fit the measured dynamic Web object size distribution for one case ofan entertainment site [11]. Note that the Weibull distribution in this case is not a long-taileddistribution because the shape parameter ν is not smaller than . The mean message size ℓ ( m ) is . bytes, and the standard deviation σ ( m ) is . bytes. 7 Frame size [bytes] C u m u l a t i v e d i s t r i bu t i on f un c t i on Message sizeBit error freeFrame size Bit errorrate= 10 -5 -4 Payload size: 2312 bytesRetry limit: 7 a) Case of static Web objects. C u m u l a t i v e d i s t r i bu t i on f un c t i on Frame size [bytes]
Message sizeBit error free Bit errorrate= 10 -5 -4 Frame sizePayload size: 2312 bytesRetry limit: 7 b) Case of dynamic Web objects.Fig. 2C
UMULATIVE FRAME SIZE DISTRIBUTIONS F ( f ) ( · ) FOR DIFFERENT BIT ERROR RATES p e . -6 -5 -4 -3 Bit error rate M ean s i z e [ b y t e s ]
512 bytesPayload size = 2312 bytes1500 bytesTransffered packetRetry limit: infiniteGenerated packet a) Case of static Web objects. -6 -5 -4 -3 Bit-error rate M ean s i z e [ b y t e s ]
512 bytesPayload size = 2312 bytes1500 bytesTransffered packetRetry limit: infiniteGenerated packet b) Case of dynamic Web objects.Fig. 3M
EAN TRANSFERRED PACKET SIZE ℓ ( q ) AND MEAN GENERATED PACKET SIZE ℓ ( p ) VERSUS BIT ERROR RATES p e WHEN RETRY LIMIT n RL GOES TO INFINITE FOR DIFFERENT PAYLOAD SIZES ℓ d . A. Effect of RPSP on frame size distribution
Figures 2 (a) and (b) show the distributions of frame sizes F ( f ) ( · ) for different bit error rates p e of static and dynamic Web objects, respectively. We used payload size ℓ d of bytes andretry limit n RL of . Note that bytes of the payload size ℓ d is the maximum transmissionunit size of IEEE 802.11 wireless LANs and of retry limit n RL is the default value [1].These figures show that the frame size distribution F ( f ) ( · ) for high bit error rates is signifi-cantly different from that for bit error free. Thus, we can see that the effect of RPSP produces amore concave curve for the transferred packet size distribution when the bit error rate is higher.Let ℓ ( q ) be the mean transferred packet size, that is ℓ ( q ) △ = R ∞ x dF ( q ) ( x ) . To investigatethe effect of RPSP when retry limit n RL goes to infinite, Figs. 3 (a) and (b) show meantransferred packet size ℓ ( q ) and mean generated packet size ℓ ( p ) of static and dynamic Webobjects, respectively, versus bit error rates p e for different payload sizes ℓ d . Table II lists meansize of transferred packets ℓ ( q ) for different bit error rates p e when payload size ℓ d is 2312 byte8 ABLE IIM
EAN SIZE OF TRANSFERRED PACKETS ℓ ( q ) FOR DIFFERENT BIT ERROR RATES p e WHEN PAYLOAD SIZE ℓ d IS BYTE AND RETRY LIMIT n RL GOES TO INFINITE . p e − − − − Static Web objects . . . . Dynamic Web objects . . . . Note:
Mean sizes of transferred packets ℓ ( q ) are represented in units of bytes. Maximum size ofgenerated packets ℓ ( p )max of static and dynamic Web objects is . bytes, which is ℓ d + ℓ (R) h . and retry limit n RL goes to infinite. in the cases of static and dynamic Web objects. FromFigs. 3 (a) and (b), and Table II, we find that the RPSP effect appears when the bit errorrate p e exceeds − . The reason for this is that longer transferred packets are likely to beretransmitted more times. Letting random variables L ( p ) κ and R κ be size and the number ofretransmissions of the transferred packet of which seqNum is κ , respectively, this implies that h ( x ( p ) κ , ∞ ) = E [ R κ + 1 | L ( p ) κ = x ( p ) κ ] > h ( x ( p ) κ ′ , ∞ ) if x ( p ) κ > x ( p ) κ ′ .Let ℓ ( p )max be the maximum generated packet size, i.e., ℓ ( p )max = min { l ; F ( p ) ( l ) = 1 } . From aninspection of Figs. 3 (a) and (b), and Table II, we find that ℓ ( q ) reaches around ℓ ( p )max as p e → .This implies that the number of transmissions of the longest transferred packets is dominant inthe total number of transmissions of all transferred packets due to RPSP. Then, we have thefollowing conjecture. Conjecture 1
Asymptotic bound on mean transferred packet size.
We denote the asymptoticbound on the mean transferred packet size by ℓ ( q )max . That is the finite limit of the mean transferredpacket size as the value of p e approaches one. Then, we have ℓ ( q ) → ℓ ( q )max = ℓ ( p )max , as p e → . (23)Appendix IV provides the proof of conjecture 1 in the case of a discrete generated packetsize distribution.From conjecture 1, we find that RPSP effect appears stronger when ℓ ( p )max /ℓ ( p ) increases.If the mean message size ℓ ( m ) is enough large compared with payload size ℓ d , resulting in ℓ ( p ) ≈ ℓ d = ℓ ( p )max , RPSP effect is likely to disappear. B. Effect of RPSP on goodput
In this subsection, we investigate the RPSP effect on goodput. To do this, we introduce ˆ G which is obtained from the approximation of F ( q ) ( x ) = ( x − ℓ ( p ) ) . Thus, ˆ G = Pr . ( delivery | L ( p ) = ℓ ( p ) ) ( ℓ ( p ) − ℓ (R) h ) E (cid:2) R + 1 | L ( p ) = ℓ ( p ) (cid:3) E (cid:2) T ( cycle ) | L ( p ) = ℓ ( p ) (cid:3) . (24)Cleary, the value of ˆ G is equal to that of ˆ G when no transferred packet loss happens becauseRPSP effect disappears (see Example 3).Figures 4 (a) and (b) show G and ˆ G versus bit error rate p e for different payload sizes ℓ d when limit retry n RL is in the cases of static and dynamic Web objects, respectively. Fromthese figures, we find that RPSP leads to overestimate goodput obtained from the traditionalmodel which assume that the transferred packets is constant in size. As similar to the results9 -2 -1 -6 -5 -4 -3 G oodpu t [ M bp s ] Bit error rate(with RPSP)(without RPSP)Payload size = 512 bytes1500 bytes2312 bytesRetry limit: 7 GG a) Case of static Web objects. -2 -1 -6 -5 -4 -3 G oodpu t [ M bp s ] Bit error rateRetry limit: 7Payload size = 512 bytes1500 bytes2312 bytes(with RPSP)(without RPSP) GG b) Case of dynamic Web objects.Fig. 4G OODPUT G AND ˆ G VERSUS BIT ERROR RATE p e FOR DIFFERENT PAYLOAD SIZES ℓ d WHEN LIMIT RETRY n RL IS . -6 -5 -4 G oodpu t r e l a t i v e d i ff e r en c e Bit error rate
Retry limit [%]
Fig. 5G
OODPUT RELATIVE DIFFERENCE ( ˆ G − G ) /G VERSUS BIT ERROR RATE p e AND RETRY LIMIT n RL WHENPAYLOAD SIZE ℓ d IS BYTES IN THE CASE OF STATIC W EB OBJECTS . mentioned in the preceding subsection, we find that the RPSP effect on goodput appears whenthe bit error rate p e exceeds − and payload size ℓ d exceeds bytes.Figure 5 shows goodput relative difference ( ˆ G − G ) /G versus bit error rate p e and retry limit n RL when payload size ℓ d is 2312 bytes in the case of static Web objects. From this figure, wefind that the effect of RPSP on goodput appears stronger when bit error rate p e increases forlarge enough retry limits. VI. C
ONCLUSION
In this paper, we have described a data-unit-size distribution model to represent the re-transmitted packet size (RPSP) property and message segmentation behavior when frames are Letting ℓ ( m )max be the maximum message size, ℓ ( p )max is given by min { ℓ d , ℓ ( m )max } . − and payload size exceeds bytes in a scenariowhere static Web objects are delivered over an IEEE 802.11 DCF wireless network.The remaining issues include modeling a scenario where the collisions happen over a wirelessnetwork with bit errors occurring in burst. A CKNOWLEDGMENT
This work was supported by JSPS KAKENHI Grant Number JP15K00139. A PPENDIX IP ROOF OF P ROPOSITION First, without loss of generality, we consider the case of discrete message size distributions,resulting in the form of discrete generated packet size distributions given by (7). Substituting(7) into (11), we have F ( q ) ( x ) = n d X i =0 w ( q ) i ( x − ℓ ( p ) i ) , (25)where w ( q ) i for i = 0 , , · · · , n d is given by w ( q ) i = w ( p ) i E h R + 1 | L ( p ) = ℓ ( p ) i i E [ R + 1]= w ( p ) i E h R + 1 | L ( p ) = ℓ ( p ) i i n d X j =0 w ( p ) j E h R + 1 | L ( p ) = ℓ ( p ) j i . (26)To derive (25) and (26), we introduce the following notations of the generated packet of sizeequal to ℓ ( p ) i for i = 0 , , · · · , n d : M i ( t ) : number of attempts of transmissions of transferred packets prior to time t , Q i,κ ( t ) :number of attempts of transmissions of transferred packets that are created from thegenerated packet with seqNum of κ prior to time t .A sender transmits the transferred packet of which seqNum is κ and size is ℓ ( p ) i Q i,κ ( t ) times prior to time t . From the argument of a probability mass function, the form of w ( q ) i canbe written as w ( q ) i = lim t →∞ M i ( t ) X κ =1 Q i,κ ( t ) n d X i =0 M i ( t ) X κ =1 Q i,κ ( t ) = lim t →∞ n d X j =0 M j ( t ) n d X i =0 M i ( t ) X κ =1 Q i,κ ( t ) lim t →∞ M i ( t ) n d X j =0 M j ( t ) lim t →∞ M i ( t ) X κ =1 Q i,κ ( t ) M i ( t ) . (27)11 ig. 6T HE EXAMPLE OF { T i,j,κ } UNDER A HEAVY TRAFFIC CONDITION IN THE CASE OF n d EQUAL TO ONE . The form of w ( p ) i in (26) is given by w ( p ) i = Pr . ( the generated packet of size is equal to ℓ ( p ) i ) = lim t →∞ M i ( t ) n d X j =0 M j ( t ) . (28)Let R κ be the number of retransmissions of the generated packet of which seqNum is κ . Under assumption A2 , { R κ } forms a sequence of mutually independent and identicallydistributed random variables with finite value of E [ R κ ]( △ = E [ R ]) . From the Law of LargeNumbers, we have E [ R + 1] = lim t →∞ n d X i =0 M i ( t ) X κ =1 Q i,κ ( t ) n d X j =0 M j ( t ) , (29)and E [ R + 1 | L ( p ) = ℓ ( p ) i ] = lim t →∞ M i ( t ) X κ =1 Q i,κ ( t ) M i ( t ) . (30)Substituting (28), (29) and (30) into (27), we obtain (25) and (26).Next, we provide an alternative derivation of (11). Consider a packet size sequence { L ( q ) n ; n ∈N ( △ = { , , · · · } ) } where L ( q ) n means the transferred packet size of the n th transmission.Forming transferred packets with the same seqNum a group, we constitute a sequence { L ( q ) n } expressed as { L ( q ) n ; n ∈ N } = n R +1 z }| { L ( p )1 , · · · , L ( p )1 , L ( p )1 , R +1 z }| { L ( p )2 , · · · , L ( p )2 , L ( p )2 , · · · , R κ +1 z }| { L ( p ) κ , · · · , L ( p ) κ , L ( p ) κ , · · · o . (31)As shown in (31), the random variable L ( p ) κ appears R κ + 1 times consecutively in the sequenceof the transferred packets with seqNum of κ . Therefore, we obtain (11).12 PPENDIX
IIP
ROOF OF P ROPOSITION Similar to the proof mentioned in Appendix I, we consider the case of discrete message sizedistributions given by (7). Substituting (7) into (16), we have G = n d X i =0 w ( p ) i Pr . ( delivery | L ( p ) = ℓ ( p ) i ) (cid:16) ℓ ( p ) i − ℓ (R) h (cid:17) n d X i =0 w ( p ) i E h R + 1 | L ( p ) = ℓ ( p ) i i E h T ( cycle ) | L ( p ) = ℓ ( p ) i i . (32)To derive (32), we introduce the following additional notations for the generated packet ofsize equal to ℓ ( p ) i for i = 0 , , · · · , n d : N i ( t ) : number of successful transmissions of transferred packets prior to time t , T i,j,κ : transmission of the j ( ≤ Q i,κ ( t )) th attempt for the transferred packet of which seqNum is κ ( ≤ M i ( t )) . The example of { T i,j,κ } under a heavy traffic condition in the case of n d equal to one is shown in Fig. 6.For large enough t , we have t ≈ n d X i =0 M i ( t ) X κ =1 Q i,κ ( t ) X j =1 T i,j,κ . (33)The definition of goodput yields G = lim t →∞ n d X i =0 N i ( t ) (cid:16) ℓ ( p ) i − ℓ (R) h (cid:17) t . (34)Substituting (33) into (34), we have G = lim t →∞ n d X i =0 N i ( t ) (cid:16) ℓ ( p ) i − ℓ (R) h (cid:17) n d X i =0 Q i,κ ( t ) X j =1 M κ ( t ) X κ =1 T i,j,κ = lim t →∞ n d X i =0 N i ( t ) (cid:16) ℓ ( p ) i − ℓ (R) h (cid:17) n d X i =0 M i ( t ) X κ =1 Q i,κ ( t ) lim t →∞ n d X i =0 M κ ( t ) X κ =1 Q i,κ ( t ) n d X i =0 Q i,κ ( t ) X j =1 M i ( t ) X κ =1 T i,j,κ . (35)The form of Pr . ( delivery | L ( p ) = ℓ ( p ) i ) is given by Pr . ( delivery | L ( p ) = ℓ ( p ) i ) = lim t →∞ N i ( t ) M i ( t ) . (36)From (28), (29) and (36), we have lim t →∞ N i ( t ) n d X i =0 M i ( t ) X κ =1 Q i,κ ( t ) = lim t →∞ n d X j =0 M j ( t ) n d X i =0 M i ( t ) X κ =1 Q i,κ ( t ) lim t →∞ M i ( t ) n d X j =0 M j ( t ) lim t →∞ N i ( t ) M i ( t ) ! = 1 E [ R + 1] · w ( p ) i · Pr . ( delivery | L ( p ) = ℓ ( p ) i ) . (37)13he first term of (35) can be rewritten as lim t →∞ n d X i =0 N i ( t ) (cid:16) ℓ ( p ) i − ℓ (R) h (cid:17) n d X i =0 M i ( t ) X κ =1 Q i,κ ( t ) = n d X i =0 w ( p ) i Pr . ( delivery | L ( p ) = ℓ ( p ) i ) (cid:16) ℓ ( p ) i − ℓ (R) h (cid:17) E [ R + 1] . (38)Under the assumption of A2 , { T i, , , T i, , , · · · , T i, , , T i, , , · · · } forms a sequence of mutu-ally independent and identically distibuted random variables with a common distribution withmean E [ T ( cycle ) | L ( p ) = ℓ ( p ) i ] . From the Law of the Large Numbers, we have lim t →∞ M i ( t ) X κ =1 Q i,κ ( t ) X j =1 T i,j,κM i ( t ) X κ =1 Q i,κ ( t ) = E h T ( cycle ) | L ( p ) = ℓ ( p ) i i . (39)The inverse of the last term of (35) can be rewritten as lim t →∞ n d X i =0 M i ( t ) X κ =1 Q i,κ ( t ) X j =1 T i,j,κn d X i =0 M i ( t ) X κ =1 Q i,κ ( t )= n d X i =0 lim t →∞ n d X j =0 M j ( t ) n d X i =0 M i ( t ) X κ =1 Q i,κ ( t ) lim t →∞ M i ( t ) n d X j =0 M j ( t ) lim t →∞ M i ( t ) X κ =1 Q i,κ ( t ) M i ( t ) lim t →∞ M i ( t ) X κ =1 Q i,κ ( t ) X j =1 T i,j,κM i ( t ) X κ =1 Q i,κ ( t ) (40)Substituting (28), (29), (30) and (39) into the above equation, we have lim t →∞ n d X i =0 M i ( t ) X κ =1 Q i,κ ( t ) X j =1 T i,j,κn d X i =0 M i ( t ) X κ =1 Q i,κ ( t ) = n d X i =0 w ( p ) i E h R + 1 | L ( p ) = ℓ ( p ) i i E h T ( cycle ) | L ( p ) = ℓ ( p ) i i E [ R + 1] . (41)Substitution of (38) and (41) into (35) yields (32). A PPENDIX
IIID
ERIVATION OF (18)
Because no collision occurs, from the argument of [6], we have E h T ( cycle ) | L ( p ) = x i = (1 − τ ( x )) στ ( x ) + (1 − g ( x )) t suc ( x ) + g ( x ) t bit ( x ) , (42)14here τ ( x ) is the probability that a sender can transmit a transferred packet of size equal to x .From the argument of [12], τ ( x ) is given by τ ( x ) = 11 + 1 − g ( x )1 − { g ( x ) } n RL +1 n RL X r =0 b r { g ( x ) } r . (43)Substitution (43) into (42), we obtain (18). A PPENDIX
IVP
ROOF OF C ONJECTURE Suppose that the generated packets sizes follow the discrete distribution given by (7). Bysubstitution of (7) into (11), the transferred packet size distribution F ( q ) ( · ) is given by F ( q ) ( x ) △ = n d X i =0 w ( q ) i ( x − ℓ ( q ) i ) , (44)where w ( q ) i = w ( p ) i h ( ℓ ( p ) i , n RL ) n d X j =0 w ( p ) j h ( ℓ ( p ) j , n RL ) , i = 0 , , · · · , n d , (45)because E [ R + 1 | L ( p ) = ℓ ( p ) i ] = h ( ℓ ( p ) i , n RL ) and E [ R + 1] = P n d j =0 w ( p ) j h ( ℓ ( p ) j , n RL ) .Let i max be the index corresponding to the maximum generated packet size ℓ ( p )max . Thus, i max = arg max i ∈{ , , ··· ,n d } { ℓ ( p ) i } . (46)We let ¯ w ( q ) i be a finite limit of the weight corresponding to discrete transferred packet size ℓ ( p ) i as p e → and n RL → ∞ for i = 0 , , · · · , n d . From lim n RL →∞ h ( x, n RL ) = 1 / (1 − g ( x )) =1 / (1 − p e ) x + ℓ ( L ) h if ≤ g ( x ) < , we have ¯ w ( q ) i = lim p e → lim n RL →∞ w ( p ) i h ( ℓ ( p ) i , n RL ) n d X j =0 w ( p ) j h ( ℓ ( p ) j , n RL ) = lim p e → w ( p ) i (1 − p e ) ℓ ( p ) i + ℓ ( L ) h n d X j =0 w ( p ) j (1 − p e ) ℓ ( p ) j + ℓ ( L ) h = lim p e → w ( p ) i (1 − p e ) ℓ ( p ) i + ℓ ( L ) h (1 − p e ) ℓ ( p )max + ℓ ( L ) h n d X j =0 w ( p ) j (1 − p e ) ℓ ( p )max − ℓ ( p ) j = lim p e → w ( p ) i (1 − p e ) ℓ ( p ) i + ℓ ( L ) h (1 − p e ) ℓ ( p )max + ℓ ( L ) h n d X j =0 ,j = i max w ( p ) j (1 − p e ) ℓ ( p )max − ℓ ( p ) j + w ( p )max = ( , for i = i max , for i = i max . (47)Thus, we have F ( q ) ( x ) → ( x − ℓ ( p )max ) , as p e → . (48)Therefore, we obtain (23). 15 EFERENCES [1]
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