Delay models in Data Networks Chapter 3 Delay models in Data Networks

Section 3.2 Little`s Theorem

3.2 Little`s Theorem : average number of customers in system : mean arrival rate T:mean time a customer spends in system

Little`s Theorem Proof N(t) = number of customers in system at time t (t) = number of customers who arrived in interval [0,t] Ti = time spent in system by the i-th customer

Little`s Theorem

Little`s Theorem

3.2.3 Application of Little`s Theorem Ex3.1  : arrival rate in a transmission line NQ : average number of packets waiting in queue W : average waiting time spent by a packet in queue NQ = W

Application of Little`s Theorem If = average Tx time   =   : Average number of packets under Tx I.e. fraction of time that s busy utilization factor

Application of Little`s Theorem Ex3.2 N : average number packets in network T : average delay per packet also Ti : average delay of packets arriving at node i

3.3 M/M/1 Queuing System M/M/1 First M : arrival , Poisson Second M : service , Exponential 1 : server number

M/M/1 Queuing System Arrival Poisson process A(t) : number of arrivals from 0 to time t Number of arrivals that occur in disjoint intervals are independent Number of arrivals in any interval of length  is Poisson distributed with parameter  ,

M/M/1 Queuing System Properties of Poisson process Inter arrival times are independent and exponentially distributed with parameter  tn : time of the n-th arrival

M/M/1 Queuing System For every t0, 0

M/M/1 Queuing System A = A1+A2++AK is also Poisson with rate  = 1+ 2++ K Poisson A1 merge A2 ……….. AK

M/M/1 Queuing System P  1-P Also Poisson with P split Poisson

M/M/1 Queuing System Service time : Exponential distribution with parameter  Sn : service time of n-th customer

M/M/1 Queuing System Properties of Exponential : memoryless

Markov chain formulation Let's focus at the times,0,,2,…,k,… Nk = number of customers in system at time k = N(k) Where N(t) is continuous-time Markov Chain Nk is discrete-time Let Pij : transition probabilities = P{Nk+1=j|Nk=i}

Markov chain formulation

Markov chain formulation Note During any time interval, the total number of transitions from state n to n+1 must differ from the total number of transitions from n+1 to n by at most 1 I.e. frequency of transitions from n+1 to n = frequency of transitions from n to n+1

Markov chain formulation

Markov chain formulation Take ->0  Pn=Pn+1 Pn+1=Pn, n=0, 1, …等比數列 where = /  utilization  Pn+1=  n+1P0, n=0,1,… Since <1, and

Markov chain formulation

M/G/1 System Let Ci ： customer I Wi = waiting time of Ci Xi = service time of Ci Ni = # of customers found waiting in queue when Ci arrives Ri = residual service time of the customer in service when Ci arrives

M/G/1 System Ri Xi- Ni Xi-1 Ci start service Ni Ci arrives In steady-state,

M/G/1 System To calculate R, by graphical approach: Residual service time r() M(t)=# of service completion in [0, t] X1 X2 XM(t) Time  X2 X1 XM(t) Ci starts service t

M/G/1 System Time avg of r() in [0, t]

M/G/1 System P-K Formula (3.53)

Ex3.15 Consider a go back n ARQ: sender 1 2 3 … n-1 n 1 time Timeout (n-1) frames 1 2 3 receiver time Prop. delay Assume that error in the forward channel is p, return channel is error-free Packet arrives as a Poisson process with rate  packets/frame

Ex3.15 Service time X : from when a packet transmitted until it is successfully received 1 , if 1st tx is successful (1-p) X={ 1+n, if 1st tx is un- successful; 2nd is successful  p(1-p) 1+kn, if 1st k is un- successful;(k+1)th successful  Pk(1-p)

Ex3.15

Ex3.15

3.5.1 M/G/1 Queue with vacations When the server has served all customers, it goes on vacation If the system is still idle after a vacation interval, go on another vacation interval If a customer arrives during a vacation, customer waits until the end of vacation. Chapter 1 section 1.3.1 page 34in Network or Transport Layer

M/G/1 Queue with vacations

M/G/1 Queue with vacations Assume vacation intervals v1, v2… are iid and are independent of customers arrival & service times. →A customer must wait for the completion of the current service or vacation interval, and then the service of all customers waiting before it.

M/G/1 Queue with vacations Where R is the mean residual time for completion of service or vacation when the customer arrives.

M/G/1 Queue with vacations Let L(t) = # of vacations completed by t M(t) = # of services completed by t

Because Fraction of time occupied with vacation = 1-

Ex3.16 : FDM, SFDM, TDM m streams of traffic with rate /m(Poisson) FDM system – Divide available bandwidth into m subchannels. Transmission time for a packet on each of these subchannels is m.

FDM

Slotted FDM System Packet trans starts only at time m, 2m,…When the queue is idle, server takes a vacation of m. (if idle again after vacation, take another)

TDM System Look at SFDM queue, ->same queue WTDM=WSFDM

Summary Service time

Reservations & Polling Satellite Collision -> solution:polling or reservation … S1 D1 D1 S2 D2 S1 D1 D1 S2 D2 Cycle

Reservation & Polling M Poisson traffic streams with rate /m Gated System – only those packets which arrive prior to the user’s preceding reservation period are transmitted. Exhaustive system – all packets are transmitted including those that arrive during this data period Partially gated – all packets that arrive up to the beginning of the data interval.

Single-User Gated system: m=1 Di arrives Di starts Wi time S D D S D D … D Di ends tx Ri Vl(I) l(i)-th reservation interval Ni : # of packets arrive in front of Di

Single-User A reservation(vacation)starts when the system has served all packets which arrive prior to the preceding reservation interval. A vacation(M/G/1 queue with vacation) starts when the system has served all packets which have arrived.(corresponds to exhaustive system)

Single-User

Single-User Single-user gated

Multi-User Packet i arrives Packet i starts Wi time S D D D … S D D … S D D Ri Pakcet i ends Ni Sum=Yi Ni is redefined as # of packets which must be transmitted before packet i

Multi-user Where Yi : includes all reservation intervals packet I must want for.

Multi-User If number the users 0, 1, 2,…,m-1, the l-th reservation interval is used to make reservation for user l mod m

Multi-user

Packet i belongs to each user with same prob. = 1/m Multi-user For an exhaustive system Let lj=E ( Yj | packet i arrives in user l’s reservation or data intervals and belongs to user (l+j) mod m) Packet i belongs to each user with same prob. = 1/m

Multi-user

Multi-user All users have equal average data length in steady state. P(packet i arrives during user l’s data interval) P(packet i arrives during user l’s reservation interval)

Multi-user (Yi|pkt i arrives in user l’s data or reser. int.) X P(pkt i arrives in user l’s reser. Or data int. )

Multi-user If Vl’s have same dist. Exhaustive system (3.69)

Multi-user - The partially gated system is the same as the exhaustive system except that if a packet arrives in its own user’s data interval (with prob. /m), it is delayed an extra cycle of reservation periods(mV) Y is increased by

Multi-user The fully gated system is the same as partially gated system except if a pkt arrives during a user’s own reservation interval (prob. (1-)/m) It is delayed by an additional mV Y is increased by

Priority Queuing N classes of customers class i arrives a Poisson process with rate I service time Each class joins a separate queue 1 Server 2

Priority Queuing Single server will server customers from the highest priority queue first Non-preemptive - a lower priority customer, once started, is allowed to finish, when a high priority customer arrives. Preemptive resume - Service for a low priority customer is interrupted when a high priority customer arrives and is resumed from the point of interruption when all higher priority customers have been served

Non-preemptive Let NQk=avg. # in queue for priority k Wk= avg. queueing time for priority k k = k/k = system utilization for priority k R = mean residual service time.

Non-preemptive Where 1W2 is the avg. # of higher priority customers that arrives while you are waiting

Non-preemptive Similarly,

Non-preemptive R=the residual time Where =2nd moment of the service time avg. over all priority

Non-preemptive 代入

Preemptive Note that Tk will not be affected by customers from class k+1 to n Work due to class 1 to k-1 who arrives when this customer is waiting (B) Unfinished work of Class 1 to k (A)

Preemptive