Lesson 7: M/G/1 Queuing Systems Analysis

Transcription

1 Slide supporting material Lesson 7: M/G/ Queuing Sstems Analsis Giovanni Giambene Queuing Theor and Telecommunications: Networks and Applications nd edition, Springer All rights reserved

2 Motivations for the Use of the M/G/ Theor The assumption of Poisson arrivals ma be reasonable since the Poisson model is a limiting condition of the binomial distribution. Man potential customers decide independentl about arriving. Each of them has a small probabilit of arriving in an particular time interval. Probabilit of one arrival in a small interval is approximatel proportional to the length of the interval. The exponential distribution for the service time is no longer a good approximation in current packet-switched networks: laer packets ma have a fixed length; files ma have a length better modeled b a heav-tailed distribution, e.g., Pareto distribution. Then, a general service time has to be considered. M/G/ theor can be used for modeling different aspects of the networks.

3 M/G/ Queues In the M/G/ theor, the arrival process is Poisson with mean arrival rate λ, but, the service time is not exponentiall distributed. The service process has some memor: if there is a request in service at a given instant, the residual service time of the request has a distribution that depends on the elapsed service time. A similar theoretical method to that of M/G/ queues can be applied to solve G/M/ ones.

4 Imbedded Markov Chains -D sstem state for M/G/ queues: S(t) = {n(t), τ(t)}. n(t): Number of requests in the sstem at instant t; τ(t): Elapsed time from the beginning of the service of the currentlserved request. To simplif the stud, the M/G/ queue is analed at imbedding instants ζ i, this is as if we take snapshots of the sstem state at instants ζ i when we obtain a mono-dimensional Markovian sstem (imbedded Markov chain), as detailed below. Different alternatives are available to select imbedding instants ζ i (especiall # and #3 below for M/G/ cases):. Service completion instants;. Customer arrival instants (used in the G/M/ case for the stud of the waiting part); 3. Regularl-spaced instants, for special cases with time-slotted service as TDM sstems (e.g., ATM): slot slot slot

5 Imbedding to Service Completion Instants Imbedding at service completion instants: τ(ζ i ) 0, i since at instant ζ i a request has completed its service and no new request has et started its service. n i denotes the number of requests in the queue soon after the service completion of the i-th request (instant ζ i+ ). a i denotes the number of requests arrived at the queue during the service time of the i-th request (ending at instant ζ i ). At instants ζ i, the state becomes mono-dimensional: S(ζ i ) n(ζ i ) = n i

6 Imbedding to Service Completion Instants (cont d) If n i 0, at the subsequent instant of service completion the following balance is valid: n i+ = n i + a i+. Note that among all requests in the queue, we do not pose special conditions on the request that has been served. If n i = 0, we have to wait for the next arrival that is immediatel served, so that at the next completion instant ζ i+ + the sstem just contains the arrivals occurred during the service time of the last request; we have: n i+ = a i+. New arrival n i 0 n i+ n j = 0 a i+ a j+ ζ i Departure instants ζ i+ ζ i+ ζ j ζ j+ time

7 Sstem Description ( ni ) a i+ = ni I + i+ where I(x) =, x > 0; I(x) = 0, x = 0 (Heaviside function). The above difference equation describes the behavior of the M/G/ queue at imbedding instants. Since the variables at the instant ζ i+ onl depend on the variables at instant ζ i, the M/G/ sstem is characteried b a discretetime Markov chain at imbedding instants ( semi-markov chain ), as shown below. n = 0 Prob{a 0 = 0} Prob{a 0 = } Prob{a = 0} Prob{a 0 = } Prob{a = } n Prob{a = 3} n = n = Prob{a = } Prob{a = } Prob{a = } Prob{a = 0} Prob{a 3 = 0} n = Prob{a 3 = }.. The arrival process is in general state-dependent. The definitions/characteristics of both n i and a i depend on the selection of imbedding instants. In general, the solution of the discretetime Markov chain (i.e., determining the state probabilit distribution) requires a matrix-geometric approach or writing cut equilibriums and an iterative solution approach. We will use an approach in the domain b adding some assumptions.

8 Solution in the -Domain with Additional Assumptions Let us assume that the M/G/ queue admits a stead state. P n denotes the probabilit (at regime) to have n requests in the queue We focus on the difference equation that is solved in the -domain (i.e., PGF) and we use the following assumptions: Memorless arrival process (a i is memorless). This is a more general condition than a Poisson process: we use the M /G/ notation, where M stands for a general memorless arrival process (e.g., a Bernoulli arrival process of packets on a slot basis). Arrival process independent of the number of requests in the queue (n i and a i are independent). This assumption is not needed using the cut equilibrium or matrix-geometric approach. ni + ni I ( ni ) ai+ Pn = + P i ni P ai+ h k j P ( )[ A( ) ] = P0 ( ) A( ) (*) where P() is the PGF of the state probabilit distribution, n i, and A() is the PGF of the number of arrivals in the service time of a request, a i.

9 Solution in the -Domain with Additional Assumptions Let us assume that the M/G/ queue admits a stead state. P n denotes the probabilit (at regime) to have n requests in the queue On both sides we take triple sum on n i+,n i, a i+ b using the joint probabilit P(n i+,n i, a i+ ). The result shown here is obtained after manipulations based on independence assumptions and marginal distributions. We focus on the difference equation that is solved in the -domain (i.e., PGF) and we use the following assumptions: Memorless arrival process (a i is memorless). This is a more general condition than a Poisson process: we use the M /G/ notation, where M stands for a general memorless arrival process (e.g., a Bernoulli arrival process of packets on a slot basis). Arrival process independent of the number of requests in the queue (n i and a i are independent). This assumption is not needed using the cut equilibrium or matrix-geometric approach. ni + ni I ( ni ) ai+ Pn = + P i ni P ai+ h k j ( )[ A( ) ] = P0 ( ) A( ) where P() is the PGF of the state probabilit distribution, n i, and A() is the PGF of the number of arrivals in the service time of a request, a i. P (*)

10 Solution in the -Domain with Additional Assumptions Let us assume that the M/G/ queue admits a stead state. P n denotes the probabilit (at regime) to have n requests in the queue We focus on the difference equation that is solved in the -domain (i.e., PGF) and we use the following assumptions: Memorless arrival process (a i is memorless). This a more general condition than a Poisson process: we service use the discipline M /G/ notation, apart the where M stands for a general memorless conditions arrival process for the (e.g., a Bernoulli arrival process of packets on a slot basis). Arrival process independent of the number of requests in the queue (n i and a i are independent). This assumption insensitivit is not needed propert. using the cut equilibrium or matrix-geometric approach. ni + ni I ( ni ) ai+ Pn = + P i ni P ai+ h k j To obtain this result we do not pose special conditions on the applicabilit of the P ( )[ A( ) ] = P0 ( ) A( ) (*) where P() is the PGF of the state probabilit distribution, n i, and A() is the PGF of the number of arrivals in the service time of a request, a i.

11 Solution in the -Domain with Additional Assumptions Let us assume that the M/G/ queue admits a stead state. P n denotes the probabilit (at regime) to have n requests in the queue We focus on the difference equation that is solved in the -domain (i.e., PGF) and we use the following assumptions: Memorless arrival process (a i is memorless). This is a more general condition than a Poisson process: we use the M /G/ notation, where M stands for a general memorless probabilit arrival process distribution (e.g., a Bernoulli arrival process of packets on a slot basis). Arrival process independent of the number of requests in the queue (n i and a i are independent). This assumption is not needed using the cut equilibrium or matrix-geometric approach. ni + ni I ( ni ) ai+ Pn = + P i ni P ai+ h k j Subscripts are here omitted because we assume to stud the at regime, that is for i. P ( )[ A( ) ] = P0 ( ) A( ) (*) where P() is the PGF of the state probabilit distribution, n i, and A() is the PGF of the number of arrivals in the service time of a request, a i.

12 Solution in the -Domain (cont d) We can derive P() as: P A( ) ( ) = P A( ) 0 In this P() formula we have an apparent singularit at =, but we can appl the Abel theorem to state that it exists the lim of P() for -pole-ero cancellation- and should be necessaril equal to for the normaliation condition. Therefore, we can solve this limit b means of the Hôpital rule: lim P0 A( ) = P0 lim = P 0 A( ) A'( ) Abel theorem + normaliation Hôpital rule = A'()

13 Solution in the -Domain (cont d) Deriving with respect to both sides of the -equation (*) and computing the result at =, at the different orders of the derivative we obtain first the empt queue probabilit P 0 and then the mean number of requests in the queue N: First derivative: P 0 = A () (normaliation condition); Second derivative: N = P' ( ) = A'() + A''() [ A'() ] The PGF of the state probabilit distribution P() onl depends on the PGF A() that, in turn, depends on the characteristics of the arrival process, the imbedding instants, and the distribution of the service time. These results are insensitive to the service discipline adopted for the queue. This solution is for a generalied queue (not onl Poisson arrivals). Stabilit condition is P 0 > 0 A () < Erl; A () is the traffic intensit.

14 Solution of the M/G/ Queue for Poisson Arrivals Assumptions: Poisson arrival process and sstem imbedded at the service completion instants. A() can be computed considering the PGF of the number of arrivals in a given interval t, A( t) = e λt( ) and then removing the conditioning b means of the probabilit densit function of the service time, g(t) [with corresponding Laplace transform Γ(s)]: A( ) or equivalentl + = e 0 λt ( ) g( t) dt = Γ( s = λ( ) ) s A = = Γ λ ( s) s to domain transform: s = λ( ) to s domain inverse transform: = s/λ

15 Solution of the M/G/ Queue for Poisson Arrivals (cont d) We obtain: A ()=λe[x] = traffic intensit ρ and A ()=λ E[X ]. Then, we can determine the mean number of requests in the sstem N as: N = A'() + A''() [ A'() ] = λe λ E [ ] [ X ] X + ( λe[ X ]) Queuing term Then, the mean dela T is obtained dividing N b λ according to the Little theorem: Service part T = N = λ E λe [ ] [ X ] X + [ λe[ X ] Pollacek-Khinchin formula

16 M/D/ Queue In this sstem, arrivals are according to a Poisson process with mean rate λ and have a fixed, constant service time, x. This is for instance the case of the transmission of packets of a given sie on a link with constant capacit. Imbedding points are at the end of the service of a request. We can directl appl the Pollacek-Khinchin formula to determine the mean dela as: T = x + λx [ λx] For completeness, we have also A() = e λx( ) and P ( ) ( ) ( ) λx( ) e λx = λx e ( )

17 M [L()] /D/ Queue This is a case with a bulk (or compound) Poisson arrival process with PGF of the message length L() in packets. The lengths of messages are iid. Each packet transmission time is here denoted b T. We are interested in determining the PGF of the number of packets in the buffer, P(), and the mean packet dela. We imbed the sstem at the end of a packet transmission. We appl the classical M/G/ theor and we need to derive A(), the PGF of the number of packets arrived in the time to serve one packet: ( ) ( ) n A n = L ' A' ( ) = λtl ( ) ( ) ( ) ( ) n n λ T λt λt ( L( ) ) A = L e = e ' A'' ( ) = λtl ( ) + λtl n n! We can write the classical M/G/ difference equation with some approximation in the case n i = 0. The mean number of packets in the sstem N p and the mean dela for the transmission of a packet T p are: N p = A'() + A''() [ A'() ] p ( ) [ s] '' [ ] ( ) T p N = ' λl The Little theorem is here applied to a compound process

18 M [L()] /D/ Queue The classical This is a case M/G/ with a difference bulk (or compound) equation Poisson arrival process with PGF of the message length L() in packets. The lengths of messages are iid. can be used as a first approximation: we Each packet transmission time is here denoted b T. consider We are n i+ interested a i+ in for determining n i = 0 (i.e., the PGF we of the number of packets in the neglect buffer, the P(), existence and the mean of the packet packets dela. after the first We imbed one in the a sstem message at the arriving end of a at packet an transmission. We appl the classical M/G/ theor and we need to derive A(), the PGF of the empt buffer). We can remove this number of packets arrived in the time to serve one packet: approximation b using the M/G/ theor ( ) ( ) with different service times, as shown in Lesson No. 9. n A n = L ' A' ( ) = λtl ( ) ( ) ( ) ( ) n n λ T λt λt ( L( ) ) A = L e = e ' A'' ( ) = λtl ( ) + λtl n n! We can write the classical M/G/ difference equation with some approximation in the case n i = 0. The mean number of packets in the sstem N p and the mean dela for the transmission of a packet T p are: N p = A'() + A''() [ A'() ] p ( ) [ s] '' [ ] ( ) T p N = ' λl The Little theorem is here applied to a compound process

19 M [L()] /D/ Queue This is a case with a bulk (or compound) Poisson arrival process with PGF of the message length L() in packets. The lengths of messages are iid. Each packet transmission time is here denoted b T. We are interested in determining the PGF of the number of packets in the buffer, P(), and the mean packet dela. We imbed the sstem at the end of a packet transmission. We appl the classical M/G/ theor These and two we need models to derive for A(), the the same PGF sstem of the number of packets arrived in the time to serve one packet: ( ) ( ) The same sstem admits another M/G/ model working at the level of messages; imbedding points are now at the end of message service times (transmissions). This is a trivial application of the Pollacek-Khinchin formula: A() = e λt( ). are both interesting: the M [L()] /D/ model characteries the sstem at the level of packets (number, dela); instead, '' [ ] the ( ) M/G/ model characteries the sstem at the level of messages (number, dela). n A n = L ' A' ( ) = λtl ( ) ( ) ( ) ( ) n n λ T λt λt ( L( ) ) A = L e = e ' A'' ( ) = λtl ( ) + λtl n n! We can write the classical M/G/ difference equation with some approximation in the case n i = 0. The mean number of packets in the sstem N p and the mean dela for the transmission of a packet T p are: A''() N p = A'() + N p [ A'() ] T s p = ' λl ( ) [ ] The Little theorem is here applied to a compound process

20 M/G/ Dela Distribution in the FIFO Case with Poisson Arrivals At service completion instant, the n requests left in the sstem Arrival instant of red are those packet arrived at the queue during the Buffer sstem dela T D experienced b a finding other packets request served. inside The probabilit distribution for n coincides with the state probabilit distribution with PGF P(). After time T D, the Being f sstem TD (t) the dela densit function of the sstem dela [T D (s) being experienced b the red the Laplace packet transform], we can write in the -domain: Buffer Completion instant of red + packet leaving λt( n ) packets P( ) e f ( t) dt = T s = λ( ) [ ] Substituting the P() expression for the M/G/ the packets queue in the queue and at using the the imbedding instant as inverse transform = s/λ, we obtain modeled the b Laplace PGF P() transform of the dela distribution: T D ( s) = P( ) ( ) A( ) A( ) = Random variable T D and PGF P() are thus related... arrived at the queue TDin the 0 meanwhile. These are also P s = Γ( s) s λ + Γ( 0 = P = s / λ 0 λ s = s / λ ) D Queuing dela

21 M/G/ Dela Distribution in the FIFO Case with Poisson Arrivals At service completion instant, the n requests left in the sstem are those arrived during the sstem dela T D experienced b a request served. The probabilit distribution for n coincides with the state probabilit distribution with PGF P(). Being f TD (t) the densit function of the sstem dela [T D (s) being the Laplace transform], we can write in the -domain: + λt( ) P( ) e f ( t) dt = T s = λ( ) [ ] Substituting the P() expression for the M/G/ queue and using the inverse transform = s/λ, we obtain the Laplace transform of the dela distribution: T D ( s) = P( ) ( ) A( ) A( ) = 0 P s = Γ( s) s λ + Γ( 0 = P = s / λ 0 λ s = s / λ T D ) D Queuing dela

22 Dela Distribution Analsis for the M [L()] /D/ Case with FIFO In the FIFO case with a bulk (compound) Poisson arrival process with PGF of the message length in packets L(), the PGF of the number of packets in the buffer, P(), and the Laplace transform of the probabilit densit function of the packet sstem dela, T Dp (s), are related b means of the condition s = λ[ L()]. If L() is the PGF of a modified geometric distribution with mean value L we have [where L (.) is the inverse function of L()]: This expression = (s) can be substituted in P() of the M/G/ solution to obtain T Dp (s) as: = L L s λ λ λ λ = = L s s s L ( ) ( ) [ ] ( ) ( ) [ ] λ λ λ λ λ λ λ s T s T D e L L s L s e s TL s T p = ' inversion

23 M/G/ Theor Generaliation Kleinrock principle (also b P. J. Burke): for queuing sstems where the state changes at most b + or (we refer here to the actual variations in the number of requests in the queue, not to what happens at imbedding points), the sstem distribution as seen b an arriving customer will be the same as that seen b a departing customer. Hence, the state probabilit distribution b imbedding the queue at the departure instants is equal to the state probabilit distributions at arrival instants. Due to the PASTA propert, the state probabilit distribution at arrival instants is valid at generic instants (random observer). The state probabilit distribution at the service completion instants coincides with the distribution of the continuoustime sstem (random observer). L. Kleinrock. Queueing Sstems. New York: Wile, 975

24 M/G/ Theor Generaliation Kleinrock principle (also b P. J. Burke): for queuing sstems where the state changes at most b + or (we refer here to the actual variations in the number of requests in the queue, not to what happens at imbedding points), the sstem distribution For a compound as seen Poisson b an arriving process customer the will be the same generaliation as that seen b considered a departing here customer. is not Hence, applicable. the state probabilit The Kleinrock distribution principle b imbedding is not the queue at the departure applicable. instants is equal to the state probabilit distributions at arrival instants. Due to the PASTA propert, the state probabilit distribution at arrival instants is valid at generic instants (random observer). The state probabilit distribution at the service completion instants coincides with the distribution of the continuoustime sstem (random observer). L. Kleinrock. Queueing Sstems. New York: Wile, 975

25 M/G/ Theor Generaliation Kleinrock In the case principle of a Bernoulli (also b P. arrival J. Burke): for queuing sstems process where on the a slot state basis changes (for which at most we b + or (we refer here to the actual variations in the number of requests in the can queue, appl the not M /G/ to what theor), happens the at imbedding points), the sstem BASTA distribution analogous as seen propert b an holds, arriving so customer that will be the we same can as reappl that seen the b generaliation a departing customer. result below. Hence, the state probabilit distribution b imbedding the queue at the departure instants is equal to the state probabilit distributions at arrival instants. Due to the PASTA propert, the state probabilit distribution at arrival instants is valid at generic instants (random observer). The state probabilit distribution at the service completion instants coincides with the distribution of the continuoustime sstem (random observer). L. Kleinrock. Queueing Sstems. New York: Wile, 975

26 M/G/ Theor Generaliation (cont d) As a further proof of the generaliation of the state probabilit distribution of M/G/ at generic instants, we could use the following heuristic considerations. The Pollacek-Khinchin formula can also be applied to the M/M/ queue (imbedding points at the service completion instants), where mean and mean square values of the service time X are so related (exponential distribution case): E[X ] = E[X]. T = E λe [ ] [ X ] X + [ λe[ X ] T = exponential service time E [ X ] + λe[ X ] [ λe[ X ] = [ X ] λe[ X ] + λe[ X ] [ λe[ X ] E = λ [ X ] E[ X ] We note that we obtain again the classical M/M/ result that is valid at an instant, not onl at imbedding points. E classical M/M/ result

27 Numerical Inversion Method for P() The PGF P() of an M/G/ queue has tpicall an expression that cannot be inverted to obtain the state probabilit distribution. A numerical inversion method is needed. As explained in Lesson No. 3, P() can be seen as a Talor series expansion centered at = 0 (i.e., MacLaurin series expansion). Hence, a simple inversion method can be obtained looking at the definition of P() : Prob d k! d { X = k} = P( ) k k = 0 This method can be easil implemented in Matlab as shown in Lesson No. 9.

28 M/G/ Theor and Heav-Tailed- Distributed Service Times Heav-tailed (Pareto) distributions for the service time are frequent in modern traffic. One disadvantage of using these distributions is that their Laplace transforms often have no closed-form expressions and are thus not eas to manipulate. The M/G/ state probabilit distribution depends on A(), the PGF of the number of arrivals in a service time. Moreover, the mean dela is given b the Pollacek-Khinchin formula, which requires to use mean and mean square values of the service time. With heav-tailed distributions, we can have infinite mean and/or variance, which ma entail some paradoxical situations for the queues, as discussed below referring to the Pareto distribution case with shape parameter γ. In the M/Pareto/ case, we need to have a finite mean value of the Pareto service time (thus entailing γ > ) in order to have a stable queue.

29 M/Pareto/ Queue If < γ, the Pareto service time has finite mean and infinite variance (i.e., heav tails). This entails that the queue is stable (there exists the state probabilit distribution as well as the distribution of the dela), but the mean dela is infinite. Hence, this is a ver special (degenerate) case, where the infinite mean dela does not impl the instabilit of the queue! The PGF of the state probabilit distribution, P(), depends on A() computed as follows: + n γ + s= λ ( ) ( ) ( ) + n λt λt γk γ λt( ) γ γ st γ A = e dt = γk e t dt k e t dt + n t = γ γ n= 0! k k k The integral in A() cannot be expressed in a closed form. It can be represented b means of the incomplete Gamma function, Γ(a, ): A ( ) = γk + γ k e st t γ dt s= λ ( ) = γ γ ( sk) Γ( γ, sk) s= λ ( ), where Γ + t a ( a, ) = e t dt If γ >, the Pareto distribution has finite mean and finite variance so that the mean dela is finite. In this case, the Pareto distribution is not heav-tailed.

30 M/G/ Mean Number of Requests for Different Serv. Time Distrib. Let us compare the mean dela of an M/G/ queue for different service distributions with the same mean arrival rate λ and mean service time E[x]. Let ρ = λe[x] < Erl denote the traffic intensit. The different service time distributions are characteried b the coefficient of variation C v : Var[ X ] C. The exponential distribution has C v =. v = E X The coefficient of variation C v is 0 for a deterministic random variable, is for an exponential distribution, is greater than for the hper-exponential distribution, and tends to for heav-tailed distributions. N M/M/ Let us compare the mean number of requests in the sstem for exponential and general service times (i.e., M/M/ vs. M/G/): λe = λ [ X ] E[ X ] We have: [ ] vs. N N M/G/ = λe λ E [ ] [ X ] X + ( λe[ X ]) Var E [ X ] [ X ] M/M/ < NM/G/ Cv = > = λe [ X ] ( λe[ X ]) ( C + ) ( λe[ X ]) = N + λe N + [ X ] v C v M/M/ Var E [ X ] [ X ] M/M/ > NM/G/ Cv = <

31 Comparison. (cont d) mean number of requests in the sstem Comparison of M/G/ with different distributions Weibull k = 0. (C v = 5) Weibull k = 0.5 (C v = 5) Exponential (Weibull k =, C v = ) Raleigh (Weibull k =, C v = 0.6) Deterministic (C v = 0) C v traffic intensit, ρ [Erl] Weibull distribution: f k k t k t β, k β ( t) = e, t 0 β β The Weibull distribution is used since varing parameter k, we can obtain distributions with different C v values from low values (< ) to high values (> ). At a parit of ρ, the mean waiting time of the M/G/ queue increases with C v, the square coefficient of variation of the service time.

32 First Exercises on M/G/ Theor

33 Exercise # We have a buffer of a transmission line that receives messages coming from two independent processes: First traffic: Poisson message arrival process with mean rate λ and exponentiall-distributed service time with mean rate µ ; Second traffic: Poisson message arrival process with mean rate λ and exponentiall-distributed service time with mean rate µ. Assuming µ µ, we have to determine the mean dela from the message arrival (total arrival process sum of both processes) to the buffer to its transmission completion.

34 Solution of Exercise The first and the second arrival processes are at the input of the buffer. Since the are independent Poisson processes, their sum is still Poisson with mean rate λ + λ. The service time probabilit densit function, f(t), is not exponential; it can be derived as weighted sum of the probabilit densit functions related to the two different input flows: λ λ + λ µ t µ t ( t) = µ e + µ e f λ λ + λ We model the buffer b means of an M/G/ queue: we imbed the chain at the instants of message transmission completion and we use the Pollacek-Khinchin formula. λ λ [ ] ( ) [ ] E[ X ] = + λ + λ E X T = E X + λ + λ µ λ + λ µ [ ( λ + λ ) E[ X ] where λ λ E[ X ] = + λ + λ ( µ ) λ + λ ( µ ) Stabilit: (λ + λ )E[X] = λ /µ + λ /µ < Erl The intensities of the two traffic flows sum. Hper-exponential service time distribution (C v > )

35 Exercise # We consider a link with a transmission buffer where messages arrive according to a Poisson process with mean arrival rate λ. Each message is formed of a random number of packets, each requiring a time T to be transmitted (compound Poisson process). L() denotes the PGF of the message length in packets that also corresponds to the PGF of the message transmission time in T units. Note: All the packets of the same message arrive simultaneousl. The arrival process and the transmission one are continuoustime (non-time-slotted). It is requested to determine the mean message dela for a generic L() b selecting suitable imbedding instants.

36 Exercise # The arrival process at the packet level is compound We Poisson; consider instead, a link with the a same transmission buffer where messages arrive according to a Poisson process with mean arrival rate λ. arrival process is simpl Each Poisson message at the is message formed of level. a random number of packets, each requiring a time T to be transmitted (compound Poisson process). L() denotes the PGF of the message length in packets that also corresponds to the PGF of the message transmission time in T units. Note: All the packets of the same message arrive simultaneousl. The arrival process and the transmission one are continuoustime (non-time-slotted). It is requested to determine the mean message dela for a generic L() b selecting suitable imbedding instants.

37 Solution of Exercise Let us imbed the sstem at the instants of message transmission completion: this is the best option to measure the performance at the message level (imbedding at the end of packet transmission is not suitable to determine the mean message dela). Let n i represent the number of messages in the buffer at the end of the transmission of the i-th message; let a i denote the number of messages arrived at the buffer during the service time of the i-th message. We have a classical M/G/ queue with Poisson arrival process. Then, we directl appl the Pollacek-Khinchin formula to derive the mean message dela: Mean value of the message transmission time T m = L' ( ) λ T + [ T ] [ L'' ( ) + L' ( ) ] [ L' ( ) λ T ] Mean square value of the message transmission time [ seconds] The stabilit condition is λtl () < Erl

38 Exercise #3 (ATM-like case, M /D/ queue) Let us consider that fixed-sie packets arrive at a transmission buffer from two TDM input lines: line # and line #. The transmission of packets from the buffer is according to a TDM output line. Input and output slots have the same duration. Input TDM lines are snchronous each other and snchronous with the output line as well. A slot of the input line # carries a packet with probabilit p; a slot of the input line # carries a packet with probabilit q. The arrival processes on the two lines are memorless and independent. It is requested to determine the mean dela that a packet experiences from the arrival at the buffer to the end of its transmission. This is a first example of discrete-time sstem.

39 Solution of Exercise 3 Each input line contributes a Bernoulli arrival process on a slot basis. The resulting input process at the buffer is Binomial on a slot basis. TDM line # + TDM line # Input lines Buffer Output line We stud this discrete-time sstem b imbedding at the end of the slots of the output TDM line. Let n i denote the number of packets in the buffer at the end of the i-th slot. Let a i denote the number of packets arrived from the two input lines in the buffer during the i-th slot (we consider here the sum of the independent input processes from lines # and #). We can write the following balance: n i+ = n i +a i+ for n i > 0 and n i+ = a i+ for n i = 0. This is the classical difference equation of M/G/ sstems.

40 Solution of Exercise 3 Each input line contributes a Bernoulli arrival process on a slot basis. The resulting input process at the buffer is Binomial on a slot basis. TDM line # + TDM line # We stud this discrete-time sstem b imbedding at the end of the slots of the output TDM line. We consider a classical assumption for this tpe of Let sstems: n i denote a packet the number must have of packets in the buffer at the end of the completel i-th slot. arrived Let a( i denote slot) before the number its of packets arrived from the two transmission input lines can in start, the according buffer during to the storeand-forward sum of approach. the independent input processes from lines # and i-th slot (we consider here the #). Input lines Buffer Output line We can write the following balance: n i+ = n i +a i+ for n i > 0 and n i+ = a i+ for n i = 0. This is the classical difference equation of M/G/ sstems.

41 Solution of Exercise 3 Each input line contributes a Bernoulli arrival process on a slot basis. The resulting input process at the buffer is Binomial on a slot basis. TDM line # + TDM line # Case (a) Input lines Buffer i+-th slot We stud n i this 0 discrete-time sstem b imbedding at the end of ζ ζ the slots of the output ζ TDM line. i ζ i+ Let n i denote the number of packets in the buffer at the end of the New arrivals completing at i-th slot. Let a i denote the number of a i+ packets the end of the arrived i+-th slot from the two input lines in the buffer during the i-th slot (we No service consider completion here at the the end of the i+-th slot sum of the independent input n j = processes 0 from lines # and #). Case (b) New arrivals completing at the end of the i+-th slot We can write n i = 0 the following balance: n i+ = n i +a i+ for n i > 0 and ζ n i+ = a i+ for n i = ζ 0. This ζ is the classical i ζ difference i+ equation of M/G/ sstems. n i 0 n i+ = n i -+ a i+ a i+ n i = 0 i+-th slot Service Output completion line of a packet at the end of the i+-th slot n i+ = a i+

42 Solution of Exercise 3 (cont d) The mean number of packets in the buffer is: N c = A'() + A''() [ A'() ] [ pkts] where A() is related to the sum of two independent processes (product in the -domain of the PGFs): A ' A ( ) = ( p + p)( q + q) = ( p)( q) + [ p( q) + q( p) ] = A 0 + A + '' ( ) = A + A = p + q A ( ) = A = pq The stabilit condition is A () = p + q < Erlang A pq = The mean packet dela is derived from N b using the Little theorem: we divide N b is A (), the mean number of packets arrived per slot. + For time-slotted sstems, we consider the PGF of the number of arrivals in a slot A(). Then, A () represents the mean number of arrivals per slot. Therefore, we can appl the Little theorem dividing the mean number of requests b A (): T = N/A () is expressed in slot units.

43 Solution of Exercise 3 (cont d) The mean packet dela is: T = N A' () = + A''() A'() = + pq [ A'() ] ( p + q)( p q) [ slots]

44 Further Application Examples of the M/G/ Theor to Telecommunications

45 ARQ Scheme for Reliable Transmissions on a Link

46 Analsis of an ARQ Scheme We consider a transmission sstem with a buffer. The transmitter is used to send packets on a radio channel. We know that: Packets arrive in groups of messages (bulk arrival process) Messages arrive according to exponentiall-distributed intervals with mean value equal to T a in seconds. The length l m of a message in packets is according to the following distribution (uncorrelated from message to message): n Prob{ lm = n pkts} = q( q), n {,,...} The buffer has infinite capacit. The radio channel causes a packet to be erroneousl received with probabilit p; packet errors are uncorrelated from packet to packet. An ARQ scheme is adopted. Round trip propagation delas to receive ACKs are negligible with respect to the deterministic packet transmission time, T (note *) A packet sojourns in the buffer until its ACK is received. We have to determine the mean number of packets in the buffer and the mean dela that a packet experiences from its arrival at the buffer to its last and successful transmission. (*) The extension of this stud to a case with high propagation delas is straightforward in the ARQ stop-andwait case.

47 Analsis of an ARQ Scheme: A Model with Feedback Transmitter Receiver λ = /T a messages/s Packet arrival process Buffer Radio channel that introduces random packet errors with probabilit p and negligible (*) propagation delas Error check No error ACK Bulk arrival process: Messages arrive according to a Poisson process with mean rate λ = /T a [messages/s]. Each message contains a number of packets with modified geometric distribution with parameter q; /q = mean length of a message in packets. Service process: Due to the errors introduced b the channel, each packet requires a modified geometricall distributed number of slots (with parameter p) to be transmitted; /( p) = mean time in slot units to successfull transmit a packet. Each slot has duration T. (*) The extension of this stud to a case with high propagation delas is straightforward in the ARQ stop-andwait case.

48 Solution The queue notation is also related to the tpe of imbedding instants selected. The arrival process is compound Poisson, but we can still use the M/G/ theor. In this case, we have an M [Geom] /Geom/ sstem. We imbed the chain to the instants of successful packet transmission (i.e., without error); a packet could be transmitted man times to achieve a successful deliver. We can write as a first approximation the classical M/G/ difference equation with n i and a i. The details of this approximation (related to the bulk arrival process) will be clarified in Lesson No. 9. A() denotes the PGF of the number of packets arrived in the time required to successfull transmit a packet, T s. In the derivation of A() three random variables need to be taken into account: Number of messages arrived in T; Number of packets conveed b each message; Time necessar in slots to successfull transmit a packet b means of ARQ (neglecting the round trip propagation dela, all the ARQ schemes are almost equivalent), T s. q PGF of the message length in packets L( ) = ( q) PGF of the time in slot to successfull transmit a packet The input traffic intensit is: λt p q ( ) T s ( p) = p

49 Solution The arrival process is compound Poisson, but we can still use the M/G/ theor. In this case, we have an M [Geom] /Geom/ sstem. We imbed the chain to the instants of successful packet transmission (i.e., without error); a packet could be transmitted man times to achieve a successful deliver. We can write Note as that a first we approximation could even imbed the classical the queue M/G/ at difference equation with n i and the a i. The end details of message of this approximation transmissions (related thus to the bulk arrival process) will be obtaining clarified in a queue Lesson of No. the 9. M/Geom/, where the A() denotes the PGF of the number Geom of service packets time arrived is the in result the time of the required geometric to successfull transmit a packet, number T s. In the of derivation packets per of message A() three composed random with variables need to be taken into the geometric account: distribution of the packet service Number of messages arrived in T; time in slots. The mean message dela is thus Number of packets conveed b each given message; b the Pollacek-Khinchin formula. N.B. Time necessar in slots to successfull transmit a packet b means of ARQ (neglecting the round trip propagation dela, all the The ARQ composition schemes are almost of two equivalent), random T q s. variables with modified PGF of the message length in packets Lgeometric ( ) = distributions has still a modified geometric ( distribution q) with mean value PGF of the time in slot to successfull given b transmit the product a packet of the ( ) ( p) T s mean = values of the composing geometric variables. p The input traffic intensit is: λt p q

50 Solution (cont d) The derivatives of this compound function A() can be obtained b leaving T s () and L() in implicit forms, because this allows easier derivatives b using T s () =, T s () =/( p), L() =, L () = /q. The PGF of the number of packets arrived in the time to serve one packet, A(), is obtained b considering the twofold composition of PGFs: A' ( ) = λt λt ( L ) A = T s e p q A'' ( ) λt = q ( p) The buffer stabilit is assured if A ( = ) < Erlang λt/[q( p)] < Erlang. The mean number of packets in the ARQ sender buffer N p and the mean dela for the correct transmission of a packet T p are: N p [ ] ( ) ( ) This is the PGF of the number of packets arrived in T. A''() p b means of the Little theorem λt + + p p q qn λ = A'() + p T [ A'() ] p = [ s] ( q)

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