Cyclic%20Multiple%20Queue%20Systems%20with%20Two%20Priority%20Classes%20and%20Exhaustive%20Service

1
Cyclic Multiple Queue Systems with Two Priority
Classes and Exhaustive Service

• Author
• Jacqueline GIANINI, Faculty of Administration,
  University of Ottawa, Ottawn, Canada
• David MANFIELD, Bell-Northern Research, Ottawa,
  Canada
• Source Data Communication and System
  Performance, 1987 p511526
• Review Performance Evaluation (8) 1988 Vol.2
  April 93-115
• Presented by Ming Yuan Lin r92725034

2
Outline

• Introduction
• Background
• Two priority model
• Analysis of the exhaustive service system
• Numerical results

3
Introduction

• Cyclic multiple queue polling model arise
  naturally in computer communication networks and
  in control structures of distributed switching
  systems 1-3. This is because Round-Rubin type
  polling is a good/fair way to guarantee access to
  the interconnection medium for multiple
  distributed processor.
• For further requirements, for example, fault
  recovery or routing update traffic need to be
  transmitted with higher priority than regular
  data traffic in a large distributed packet
  switching system.

4
Introduction

• Multiple classes of priority add another level of
  complexity to the service discipline imposed by
  the Round-Rubin polling of station. (See
  figure-1)
• Service discipline
• Within a given priority level, each station is
  sequentially (cyclic) polled for messages.
• Switching overhead of changing between station i
  and i1 due to access token propagation time and
  bookkeeping operations of station i
• High priority polling (for only high priority
  messages)
• Low priority polling (for both high and low
  priority messages)

5
Introduction
6
Introduction-exhaustive service

• Gated service all stream i arrivals during
  reservation or service interval for i are queued
  until the next stream i service interval.
• Partially gated service all stream i arrivals
  during reservation period for i get transmitted,
  but arrivals during service interval for i are
  queued for the next stream i interval.
• Exhaustive service all stream i arrivals while
  the server is serving i get transmitted

7
Introduction-exhaustive service
8
Introduction

• In this paper, we consider the following system
  and evaluate the performance by mean delay.
• N stations and one single server
• Two priority queue (low priority1 and high
  priority2)
• Non-zero switching time
• High priority polling (only for high priority
  messages)
• Low priority polling (for both low and high
  priority messages)
• FCFS and exhaustive service discipline at each
  queue
• Sequential polling in a cyclic order
• Part of analysis is approximate, and simulation
  result is presented to validate the
  approximation.

9
Introduction-two priority queue
10
Outline

• Introduction
• Background
• Two priority model
• Analysis of the exhaustive service system
• Numerical results

11
Background

• The approach of this paper to the performance
  analysis of the two-priority multiqueue system
  will be based principally on previous work in
  reference 9-11.
• 9 HASHIDA, O (1972). Analysis of Multiqueue
  Rev. El. Commu. Lab., 20, 189-199
• 10 TAKAGI, H. and KLEINROCK, L. (1985).
  Analysis of Polling System, JSI Research
  Report, Computer Science
• 11 EISENBERG, M. (1972). Queues with Periodic
  Service and Changeover Times. Oper. Res., 20,
  440-451

12
Background

• In 9, the author observe that if there are n
  messages enqueued in station i at the instant of
  polling station i, then for exhaustive service
  case, the duration of the polling station i is
  the sum of n busy periods of service times at
  station i.
• From observation, it is possible to relate the
  joint moment generating function of all queue
  lengths at the instant of polling station i to
  that of the polling station i1.
• By differentiation, a set of linear equations
  yields the first two moments of the length of
  reach queue at the polling instant.

13
Background

• In 9, if the arrival process at each station is
  Poisson, queue i can be consider as an M/G/1
  queue with server vacation and yields the moment
  generating function of the length of queue i at
  message departure time.
• Then by Laplace transform, the mean delay in
  station i can be obtained as a function of the
  first two moment of the server vacation time.

14
Background
1.Moment generating function at station i
at departure time
Probability density function of mean delay
2.Calculate mean delay of messages from the first
and second moments of server vacation
3. There exist the relationship between the first
two moment of server vacation and the
expected queue length at the polling instant
15
Background -moment generating function and
Laplace transform

• See

16
Outline

• Introduction
• Background
• Two priority model
• Analysis of the exhaustive service system
• Numerical results

17
Two priority model

• In this paper, we denote the following notations.
• ?ij the arrival rate at station i, priority j
• H ij arbitrary random variable for message
  transmission time (service time)
• Mean and Variance for General Service Time r.v.
  Hij
• Busy rate for each priority queue, station and
  overall system

We assume ? lt1 to insure that the system is stable
18
Two priority model

• The switching time between station i and i1 is
  modeled by a random variable Ui and we assume the
  random variables U1,, UN are independent.
• Mean and Variance for General Switch Time r.v. Ui
• The total latency is U U1 UN, and we denote
  by u which expectation is
• Then for mean delay analysis is related to the
  study of cycle time, or time between successive
  polls of a given station.

19
Two priority model-cycle and super-cycle time
Cicycle time which is the time between
successive polling at station i
Cisuper-cycle time which is the time between
successive low priority polling at station
i
20
Two priority model-cycle and super-cycle time

• From observation, we see that the longer
  super-cycle time contains one low priority
  polling service time first the following by
  several high priority polling service time.

21
Two priority model

• In 10, the expectation of the cycle time ci can
  be shown that
• During a high priority polling with exhaustive
  service the, transmission continues at staion i
  until the high priority queue is empty. The work
  of a high priority enqueued message form a busy
  period Bi, with Laplace transform given by
  
  
• where s is the random variable of the general
  service time.
• The first two moment of Bi can be derived as

Eq. 3.1
Eq. 3.2
Eq. 3.3
22
Two priority model

• The during of a high priority polling is the sum
  of many busy period Bi of the messages in queue
  at the polling instant of station i.
• Similarly,during a low priority polling,
  transmission continue until both high and low
  priority queues are empty.
• The work of a low priority enqueued message form
  a extended busy period Bi which begins with a
  low priority message and continues with high
  priority service until the high priority queue
  becomes empty again.
• And let Si be the length of an extended service
  time at low priority (or time until the high
  priority queue becomes empty again.)

23
Two priority model

• Laplace Transform of busy period Bi with a given
  low priority extended service time r.v Si
• with Laplace Transform of a given low
  priority extended service time r.v Si
  .
• The derivation of the first and second moment of
  BI
• Again, the during of a low priority polling is
  the sum of many busy period Bi of the messages
  in low priority queue at the polling instant of
  station i.

Eq. 3.4
Eq. 3.5
Eq. 3.6
24
Outline

• Introduction
• Background
• Two priority model
• Analysis of the exhaustive service system
• Number of messages at polling instant
• Mean delay analysis
• Approximation for gi(i)
• Numerical results

25
Analysis of the exhaustive service system

• The aim of the authors is to find an
  approximation to the mean delay for both types of
  priority messages at each station.
• The authors adapt the procedure of HaShida 9 as
  presented by Takagi and Kleinrock 10, and first
  determine the joint moment generating function of
  all queue lengths at polling instant.

26
Analysis of the exhaustive service system

• Second, calculate the first two moment of the
  mean delay of messages in station i.
• Finally, because these first two moments depend
  on the expected number of low priority messages
  present at station i at a arbitrary polling
  instant of station i gi(i) which cannot be
  calculated directly, the authors approach an
  approximation to this (gi(i)).

27
Number of messages at polling instant- notation

• Li2(t) number of type 2 ( high priority)
  messages present in station i at instant t
• Li1(t) number of type 1 ( low priority)
  messages present in station i at instant t
• ti(t) n-th polling instant at station i
• Li2 number of type 2 messages present in
  station i at a arbitrary polling instant of
  station i
• Li1 number of type 1 messages present in
  station i at a arbitrary low priority polling
  instant of station i
• Fi is the joint moment generating function of
  (L12(t), L22(t),, LN2(t), L11(t), L21(t),,
  LN1(t)) at time t ti(n), where n is sufficient
  large for steady state to have been achived.

28
Number of messages at polling instant

• We can relate the Fi(x1,,xN,y1,,yN) to
  Fi1(x1,, xN,y1,,yN). Hence, by arguments
  similar to those presented in Takagi and
  Kleinrock 10, we get

Eq. 4.1
29
Number of messages at polling instant

• Then, by definition of joint moment generating
  function, we can get the mean value of the two
  priority queue lengths of station j at the
  station is polling instant
• The above is analysis at a arbitrary instant. For
  the sake of convenience to analyze the low
  priority queue length at a low priority polling
  instant, the author designed Ii0 as the indicator
  function of the event Li2(ti(t))0, j1,N and
  the probability piE(Ii0).

30
Number of messages at polling instant

• The expected queue length of station j at a
  arbitrary polling instant of station i when all
  high priority queues of stations are empty is
  defined as
• Note
• gi(j) presents the expected mean value of the
  queue length at station j at a arbitrary priority
  polling instant of station i (need approximation)
• g(0)i(j) presents the expected mean value of the
  queue length at station j at a low priority
  polling instant of station i. (solved from first
  moment linear equations)

31
Number of messages at polling instant

• And the expected number gi(j) of low priority
  messages found by the server at a arbitrary low
  polling instant of station i can be defined as
• Note

32
Number of messages at polling instant

• By differentiating equation 4.1 with respect to
  each xi and yi variables, we obtain a set of 2N2
  equations with fi(j), gi(j)and g(0)i(j),
  (i,j1,,N)

Eq. 4.2
Eq. 4.3
33
Number of messages at polling instant

• Note that we only have 2N2 equations in 2N2N2
  variables, and these equations do not suffice to
  determine a unique solution. (There exists no
  solution or infinite solution for this set of
  equations.)

34
Number of messages at polling instant

• By summing equations 4.3 respective to i, we get
• Note that the three terms in the bracket present
  switch time, high priority busy period at a
  arbitrary polling instant, and low priority busy
  period at low priority polling instant of each
  station i respectively. Then they jointly form
  the expected cycle time length Ci of station i
  (see equation 3.1)

Eq. 4.4
35
Number of messages at polling instant

• By approaching the same skill to (4.2) , these
  form a system of N2N equations with N2N
  variables fi(j) and g(0)i(j) (i,j1,,N), which
  has as it unique solution

Its stiil unable to calculate gi(i) and an
approximation to this is given in section 4.3.
36
Number of messages at polling instant

• Then, we approach the same skill to find the
  second moment of (L12(t), L22(t),, LN2(t),
  L11(t), L21(t),, LN1(t)) at time t ti(n), we
  define
• from the definition of moment generating
  functions. And we get 3N3 equations with 3N3N2
  variables in fi(j,k), hi(j,k),and g(0)i(i,j),
  (i,j,k 1,,N).

37
Number of messages at polling instant- equations
of second moments
38
Number of messages at polling instant- equations
of second moments
39
Number of messages at polling instant

• Again, by summing the above equations respect to
  i, we get

40
Number of messages at polling instant- summation
of the equations of second moments
41
Number of messages at polling instant- summation
of the equations of second moments

• with 3N3 equations and variables, which has a
  unique solution.

42
Number of messages at polling instant-symmetrical
case

• In symmetrical case, where all stations have the
  same input rate (?11?N1and?12?N2 ), service
  time distribution and switch overhead times
  distribution, we have

Eq. 4.6
The second moment of high priority queue length
of station i at an arbitrary polling instant of
station i.
43
Number of messages at polling instant-symmetrical
case

• with E(Bi2) and E(Bi2) as calculated in
  (3.3) and (3.6) respectively. But these
  expressions still contain two unknown parameters,
  pi and gi(j), the authors shall find an
  approximation to these.

Eq. 4.5
44
Outline

• Introduction
• Background
• Two priority model
• Analysis of the exhaustive service system
• Number of messages at polling instant
• Mean delay analysis
• Approximation for gi(i)
• Numerical results

45
Mean delay analysis

• To find an expression for the mean delay, we
  proceed as Eisenberg 11 and Hashida 9, and
  consider the queueing system in station i as a
  M/G/1 queue with server vacation time.

Ti service time at low priority
Ai intervisit time at low priority of station i
service time at high priority of station i
46
Mean delay analysis- low priority waiting time

• Let Si be the length of an extended service time
  at low priority and the moment generating
  function Gi1 of the low priority queue length at
  extended service completion time is given as the
  following
• Then, the Laplace transform of the waiting time
  of messages in the low priority queue is

47
Mean delay analysis- low priority waiting time

• By differentiating, we get the expectation value
  of Wi1
• Since the number of low priority messages at the
  low priority polling instant of station i is
  equal to the number of low priority message
  arrived during Ai.We can obtain the Laplace
  transform of Ai form the joint moment generating
  function.

Eq. 4.7
48
Mean delay analysis- low priority waiting time

• By differentiating, we get the first and second
  moments of Ai
• Substitute the above two to (4.7) and we get

49
Mean delay analysis- low priority polling duration

• Let us remark that the service time Ti at low
  priority is the sum of Li1 modified busy period
  Bi, and consequently the Lapalce transform can
  be obtained as
• By differentiation, we get
• From
  ,we consequently get the result
• .

Eq. 4.8
Eq. 4.9
50
Mean delay analysis- high priority waiting time

• Let us consider the high priority queue at
  station i, and we distinguish two cases
• (i) a high priority message served during a low
  priority polling of station i and let wi be the
  mean delay of these messages
• (ii) a high priority message served during a high
  priority polling of station i and let wi be the
  mean delay of these messages

51
Mean delay analysis- high priority waiting time

• The waiting time of high priority message is the
  average of that is served during low priority
  polling instant or high priority polling instant
  of the station. We denote the former as Wiand
  the later as Wi. So, the average waiting of a
  high priority message is compute by the following
  equation
  r1 and r2 present the proportion of the
  messages server during the low priority polling
  and high priority polling of station i
  respectively.

52
Mean delay analysis- high priority waiting time

• From cycle time analysis, we get

The proportion of message fall into Ti
The proportion of message fall into Ai
53
Mean delay analysis- high priority waiting time

• For Wi, the expected waiting time is related to
  the residual service time of low priority message
  (non-preemptive) and the expected high priority
  queue length that have existed before this new
  arrival.
• For Wi, the expected waiting time is related to
  the residual intervist time (A) of high priority
  queue and the expected high priority queue length
  that have existed before this new arrival.

Eq. 4.10
Eq. 4.11
54
Mean delay analysis- high priority waiting time

• By
  , we get
• To calculate E(A) and E(A2), we can relate the
  Lalpace Transform of Ai to the moment generating
  function of Li by noting that the number of high
  priority messages served at as arbitrary polling
  instant is equal to the number of high priority
  messages that arrived during Ai
• 
  where is equal to

Eq. 4.12
55
Mean delay analysis- high priority waiting time

• By differentiation, we get
• Substitute the above two to (4.12), we obtain

Eq. 4.13 4.14
56
Outline

• Introduction
• Background
• Two priority model
• Analysis of the exhaustive service system
• Number of messages at polling instant
• Mean delay analysis
• Approximation for gi(i)
• Numerical results

57
Approximation for gi(i)

• gi(i) is defined as E(Li1(t)) when t is an
  arbitrary polling instant of station i, and equal
  to the expected number of low priority queue
  messages that have arrived at station i since the
  end of the last low priority polling service of
  station i. We have
  .
• To calculated the expectation of Vi, we must look
  at the regenerative cycle for which the
  regeneration point are the instant at which all
  stations are empty for both high and low priority
  messages (the end of the low priority polling
  service).

Eq. 4.15
58
Approximation for gi(i)

• We define Xi is the number of cycles in a
  supercycle. The expectation of supercycle is
  define as
• , and since we know that
  , it follows that

59
Approximation for gi(i)

• Then by regenerative arguments, we have
• From equation (4.8), we assume Xi and Ti are
  independent and get

Eq. 4.16
Eq. 4.8
60
Approximation for gi(i)

• Then from , we obtain

Eq. 4.17
61
Approximation for gi(i)-for Xi

• Recall that Ii0 is the indicator function of the
  event Li2(ti(t))0, j1,N and the probability
  piE(Ii0). The author denote tj as the expected
  time elapsed since the last polling service of
  station j for j1,N. Then the probability that
  no high priority messages enqueued in station j
  at the instant of station i is approximated by
• The probability pi is approximated by
• By symmetric case (?12?N2) we reduce the above
  equation to

62
Approximation for gi(i)-for Xi

• To calculate tj, the expected time spent polling
  station j in an arbitrary cycle is ?ic, which in
  symmetric case is equal to for all i1,N. By the
  way, the switch time overhead with station j is
  ui which is equal to ui in symmetric case.

Eq. 4.18
63
Approximation for gi(i)-for Xi

• It would be temping now to assume that Xi is a
  geometric random variable with parameter ?i. To
  calculating the second moment of Xi, we denote qi
  as probability of polling station i at low
  priority if the previous poll was also at low
  priority (the probability of successive low
  priority polling).
• And let sk (k1,,N) be the expected duration of
  the polling of station ik during the first cycle
  of a supercycle. s0 is equal to E(Ti) and sk is
  the sum of busy periods of high priority messages
  nk?k2(s0sk-1 kui)arrived since the
  beginning of the low priority polling of station
  i.

64
Approximation for gi(i)-for Xi

• Then, the expectation of sk is
  .
• The average time spent polling station ik at low
  priority is approximately the same in cycles, and
  is consequently equal to
  we also obtain a set of recursive equations for
  sk.

65
Approximation for gi(i)-for Xi

• The same as pi, we get
• Now, we can use (4.20) and (4.19) in (4.17) to
  estimate gi(i).

Eq. 4.20
66
Approximation for gi(i)-for Xi

• To estimate the second moment of Xi, the author
  design the following auxiliary variables Xia and
  Xib which are geometric variables with
  probabilities pia and pib respectively
• Combine with the following two constraints
• We can obtain

67
Outline

• Introduction
• Background
• Two priority model
• Analysis of the exhaustive service system
• Numerical results

68
Numerical results

• Finally, the author present a few numerical
  examples with comparison to computer simulation.
• In each example, the mean delay of high/low
  priority message which are calculated in the
  above analysis are compared with the results of a
  GPSS simulation with 95 confident level.
• The error is computed by

69
Numerical results-experiment 1
The error is at acceptable level 5.
70
Numerical results-experiment 2
As the number of station becomes large and the
switching is long, low priority messages will
suffer from long delay time even in low
utilization. The reason is that the server is
always occupied by high priority messages.
71
Numerical results-experiment 3
Heavy load and unacceptable mean delay
72
Conclusion

• The authors proposed a method for analyzing
  multiqueue systems with two priority
• In this paper, the author determined the expected
  number of queue lengths of each station at the
  polling instant and the mean delay of high/low
  priority messages
• Note Linear equation solving technique and
  approximation approach