Priority queueing (nonpreemptive)

Transcription

1 Priority queueing (nonreemtive) M. Veeraraghavan, Mar. 7, 4. Mean number of jobs in a r-class PQ system with no reemtion Assume there are r classes of customers with corresonding arrival rates of λ, λ, λ, r, rankordered such that class has the highest riority and class r, the lowest. Arrival rocesses are all Poisson. Let the service rates of the r classes be denoted µ, µ, µ, r, resectively, with the service time distributions being arbitrary (general). The system is deicted in Fig.. λ Service rate: µ i for class i λ r Fig. M/G/ queue with non-reemtive riority queueing Our goal is to find the average waiting time for a class acket. EW [ + EN [ k EX [ k + k k λ k EW [ EX [ k (EQ ) V is the remaining service time for the customer in service, X k is the service time of riority-jobs, is the number of riority- k jobs in the queue. The second term reresents the waiting time incurred from having to wait until all the jobs already in queue with a riority higher than or equal to N k (since ackets that arrived before the tagged acket with rirotiy should be served before the tagged acket) are served. While the tagged job waits for service, a number of new jobs can arrive. Such jobs whose riority are wait time is reresented by the third term in (). or higher should be served before the tagged job is served. This For stability, the total system load, which is the aggregate of the er-class load ρ k ρ k r λ k EX [ k and ρ k (EQ ) k By Little s law

2 EN [ k λ k EW [ k ; (EQ 3) The traffic load σ i ρ i (EQ 4) Substituting for EN [ k from (3) into the second term in () and relacing λ k EX [ k by ρ k as er (), and using the aggregate system load from () for the third term in (), we get ( σ )EW [ + ρ i EW [ i for r (EQ 5) i ( σ ρ )EW [ + ρ i EW [ i i or (EQ 6) ( σ )EW [ + ρ i EW [ i i (EQ 7) Solving the above recursively, we get: ( ρ )EW [ (EQ 8) EW [ ρ (EQ 9) For, (7) becomes EW [ + ρ EW [ (EQ ) ( ρ ρ ) Substituting for EW [ from (9), we get

3 EW [ Generalizing the above we get: ρ + ( ρ ) ( ρ ρ ) ( ρ )( ρ ρ ) (EQ ) EW [ ( σ )( σ ) (EQ ) Exercise: Derive EW [ 3 ( σ 3 )EW [ 3 + ρ EW [ + ρ EW [ (EQ 3) ρ ( σ 3 )EW [ 3 ρ ( ρ ) ( ρ )( ρ ρ ) (EQ 4) ( σ 3 )EW [ 3 ( ρ ρ ρ ( ρ ρ ) + ρ ( ρ ρ ) + ρ ) ( ρ )( ρ ρ ) (EQ 5) EW [ ( σ 3 )( ρ ρ ) ( σ 3 )( σ ) (EQ 6) To comute, the mean residual time of the job in service: job arrival age T residual life or excess life Y X (service time) samled interval W Fig. Mean residual service time comutation Fig. shows an arbitrary service interval being samled. If the df of service time X is (service times are indeendent, identically distributed random variables), then if residual life is denoted Y (see Fig. ), then the df of Y is f X ( x) 3

4 f Y ( y) F X ( x) EX [ and (EQ 7) Average residual life EY [ EX [ EX [ (EQ 8) Above is renewal theory - see [8. An arrival either sees an idle server with robability ( ρ) (total load is ρ) because of the relation between utilization and load, or the server is occuied with robability ρ. In the former case V is. Therefore ( ρ) ρ EX [ EX [ (using (8)). (EQ 9) See M/G/ queue derivation to see if ρ EX [ EX [ λe [ X EX [ EX [ λ EX [ ( ) (EQ ) Generalizing to the case with r classes of service: r k λ k ( EX [ k ) (EQ ) Mean resonse time: ET [ EW [ µ (EQ ). Random incidence [, age 3 Say the arrival shown in Fig. occurs within some random interval. The question is whether the distribution of W, the interval in which the random incident occurred, has the same distribution as X? At first glance it aears to be so. But on closer examination, we see that in fact W and X have different distributions. This is because it is more likely for the random arrival to occur in a longer interval than a shorter one. If W w, then 4

5 f W ( w)dw wf X ( w)dw EX [ (EQ 3) The robability that the random event falls in an interval of length w is roortional to the length of that interval and to the relative occurrence of such intervals (given by f X ( w)dw); the denominator is the normalization factor. Thus the df of W is: f W ( w) wf X ( w) EX [ (EQ 4) Kleinrock uses the examle of a erson arriving at a bus terminal within the interarrival times of buses. It is more likey that a erson will arrive in a longer interarrival time than in a shorter one. If X is exonentially distributed, then f X ( x) λe λx. ( ) f W ( w) w ,. (EQ 5) λ λ we λw w > The above is the -stage Erlang distribution and EW [ λ, which means the mean length of the eriod in which the random incidence occurs is twice as long as EX [. Another way of obtaining the same result is to use the memoryless roerty. Both Y and T called the forward and backward recurrence times, resectively, will have exonential distributions with arameter λ. W is the sum of these two indeendent random variables and hence it has the Erlang distribution. To find the distribution of Y, the residual time or forward recurrence time, we use the conditional distribution: λw λe f YW ( yw) w < y w otherwise (EQ 6) For derivation of above, see [9, age 5. PT [ > t( N(, w) ) PY [ ( w t) ( N(, w) ) (EQ 7) PY [ ( w t) ( N(, w) ) P[ ( N(, t) ) ( Ntw (, ) ) PN ( (, w) ) (EQ 8) 5

6 PY [ ( w t) ( N(, w) ) e λt λ( w t) e λ( w t) λwe λw --- t w (EQ 9) In other words, if there is one occurrence in an interval, then it is uniformly located. This roerty extends to multile occurrences. But wait, does it work only if the arrival rocess is Poisson? Using (6), we get the joint df: wf, ( wy, ) f W ( w)f YW ( yw) X ( w) f X ( w) for < y w< (EQ 3) we[ X EX [ f WY f Since ( y w< ), f Y ( y) f WY, ( wy, ) dw X ( w) dw F X ( y) (same as (7)) (EQ 3) EX [ EX [ y y For the secial case of X being exonentially distributed, f Y ( y) e λy f, (EQ 3) λ X ( y) or the residual age is also exonentially distributed with the same arameter λ. For the general case, what is EY [? From [9, age 3 y( F EY [ yf Y ( y) dy X ( y) ) d EX [ y y f EX [ X ( x) dxdy y f EX [ X ( x) ydydx x (EQ 33) ( x has to be greater than y is equivalent to y having to be less than x) EY [ f EX [ X ( x) x ---- dx EX [ EX [ (EQ 34) 3. With reemtive resume riority In this system, a job can be reemted by a higher-riority customer. Resonse time consists of three terms. The first term is average service time of job itself: µ. The second term is the time to serve customers of riority through k already in the system. This is the same as the average waiting time in a non-reemtive M/G/ PQ system (customers with riorities ( k + ) through n don t mat- k T k 6

7 ter): R k λ , where R i ( EX [ i ). (EQ 35) ρ ρ ρ k k The reason for the above is that in an M/G/ queue (with a single class of customers), we derived the P-K formula as: EW [ k i EW [ λe τ [. (EQ 36) ( ρ) Reference [ writes this as equivalent to EW [ R, where (EQ 37) ρ R is the mean residual time, equal to E[ V. See (). This exlains (35). The unfinished work (sum of remaining service times of all customers in the system) of an M/G/ system is indeendent of the riority disciline of the system. This is true of all work-conserving systems (server is busy if the system is not emty). The third term is the average resonse time of customers who arrive with riority classes through k while our tagged customer of riority class k is waiting in queue or being served (reemtive). This term is equal to: Adding these three terms: k ρ i T i for k > and for k. (EQ 38) i R k k T k µ k ( ρ ρ ρ k ) ρ i Tk i (EQ 39) ( µ For k, T )( ρ ) + R (EQ 4) ρ 7

8 For k > : T k ( µ k )( ρ ρ ρ k ) + R k ( ρ ρ ρ k )( ρ ρ ρ k ) (EQ 4) (same recursive derivation as in non-reemtive case). Comare (4) with (). In (), deends uon the service time of all r classes in the system (see ()), whereas (4) deends uon R k. The definition of residual time R i seen by the i th customer is described as follows in [: if customer j is already being served when i arrives, R i is the remaining time until customer j s service time is comlete. If no customer is in service (i.e., system is emty when i arrives), is ). The mean residual time is defined as R i R lim E[ R i. (EQ 4) i Since the job in service when our tagged customer arrives may belong to any class, in the case of non-reemtive PQ deends uon the service times of all r classes. In the reemtive case, if a customer with lower riority than that of the tagged customer (arriving customer) is in service, it can be reemted. Therefore same or higher classes matter. R k aears in (4) where only the service times of the customers in the References [ D. Bertsekas and R. Gallager, Data Networks, Prentice Hall, Second Edition, 99. [ K. S. Trivedi, Probability, Statistics with Reliability, Queueing and Comuter Science Alications, First Edition, Prentice Hall, ISBN r. [3 A. Leon Garcia and I. Widjaja, Communication Networks, McGraw Hill,, First Edition. [4 E. Pinsky, A. Conway and W. Liu, Blocking Formulae for the Engset Model, IEEE Transactions on Communications, vol. 4, no. 6, June 994, [5 R. Syski, Introduction to Congestion Theory in Telehone Systems, Oliver and Boyd, Edinburgh, 96. [6 Mischa Schwartz, Telecommunications Networks, Protocols, Modeling and Analysis, Addison Wesley, 987. [7 D. Gross and C. M. Harris, Fundamentals of Queueing Theory, Wiley Series in Probability and Mathematical Statistics, 985. [8 S. M. Ross, Stochastic Processes. [9 Bob Boorstyn s notes. 8