Advanced Computer Networks Lecture 3. Models of Queuing

Transcription

1 Advanced Computer Networks Lecture 3. Models of Queuing Husheng Li Min Kao Department of Electrical Engineering and Computer Science University of Tennessee, Knoxville Spring, /13

2 Terminology of Queues A queue is represented by α/β/m, where α and β specify the types of inter-arrival time distribution and serve time distribution, respectively, and m is the number of servers. The common types of α and β include M: exponential distribution. D: deterministic GI: general independent G: general Ek : the Erlang distribution with k stages. Typical queues: M/D/1, GI/G/, M/M/1. 2/13

3 Queue Discipline FIFO: first in first out. LIFO: Last in first out. SIRO: Service in random order PS: Processor sharing RR: Round robin. 3/13

4 Metrics Q: the queue length at time t W : the actual waiting time (the time from arrival to the start of service) V : workload, the total time the servers have to work to clear the system. If there is a single server, it is the time needed for the server to clear all the queue provided that no new customers arrive. It is also called virtual waiting time. 4/13

5 GI/G/1 Queue The queue length Q(t) is given by Q(t) = x R s (Z (t)) + R a (t), where x = Q(0), R s is the serving process and R a is the arrival process. Z (t) is called the cumulative allocation process or cumulative busy time. Controlled Random Walk Model: Q(t + 1) = Q(t) S(t + 1)U(t) + A(t + 1). For M/M/1 queue, R a and R s are independent Poisson processes with parameters α and µ. 5/13

6 Sampling For M/M/1, the arrival time interval and service time are both exponentially distributed with parameters α and µ. Let R be a Poisson process with parameter α + µ. Define the renewal times: T (n) = E(1) E(n). where E is the event time of the process. If sampled process of M/M/1 at times {T (n)} can be described by the control random walk model. 6/13

7 Heavy Traffic The load of a single server queue is defined to be the ratio of the arrival (α) and service (µ) rates ρ = α. If ρ > 1, then the queue length will µ diverge. We say the queue has a heavy traffic if ρ < 1 and is very close to 1. Consider the queue length process Q k where the load is given by ρk = 1 δ. kµ The queue length process is given by Q k (t; x) = x + I k (t; x) δt + N k (t), where N k (t) can be approximated by a Gaussian random variable when k is sufficiently large. 7/13

8 Fluid Model We ignore the randomness and consider continuous queue lengths. The fluid model gives d q(t) = µξ(t) + α, dt where ξ(t) equals 1 if q(t) > 0 and 0 otherwise. 8/13

9 Klimov Model We consider a single server and l buffers, each of which is fed by one of l arrival processes. The queue length process is given by q(t) = x + Bz(t) + αt, where B = diag(µ1,..., µl ). 9/13

10 Klimov Model: Myopic Strategy Generic question: At which buffer should the server work at a given time, based on the observations of the buffer levels? The c µ Rule: The Klimov indices {i1,..., il } are defined so that cik µik cil µil whenever k l. The c µ rule for the fluid model is the policy defined so that n X ξij = 1, j=1 whenever 10/13 Pn j=1 qij > 0.

11 Processor Sharing Model We consider two servers, while server 2 can assist both queues. We can find the equivalent Klimov model. 11/13

12 Inventory Model Consider a make-to-stock production facility in which a single product is produced. The only control decisions are whether to produce or to remain idle at the station. A standard policy is based on a base-stock or hedging point x : proceed production at the fastest rate if the inventory is below x ; otherwise, stop it. 12/13

13 Power Networks As A Queue In power networks, it is a common practice for the system operator to maintain a substantial amount of reserve production capacity to meet seasonable and hourly fluctuations. We consider primary and ancillary sources of power generations. The ancillary generation is more expensive but faster. We can construct a network model for the power grid which is very similar to the inventory model. 13/13

14 Optimization in Simple Re-entrant Line The re-entrant line can be used to model networks in which there is a single arrival stream and the routing is deterministic. The buffer priority is an approach to schedule the service at each station. 14/13