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@ -6,7 +6,7 @@ Continuous-time Markov chains (CTMCs) |
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Background |
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===================== |
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In this section, we explain how Stormpy can be used to build a CTMC representing a cyclic server polling system. |
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In this section, we explain how Stormpy can be used to build a simple CTMC. |
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Building CTMCs works similar to building DTMCs as in :doc:`building_dtmcs`, but additionally every state is equipped with an exit rate. |
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.. seealso:: `01-building-ctmcs.py <todo /examples/building_ctmcs/01-building-ctmcs.py>` |
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@ -17,75 +17,50 @@ First, we import Stormpy:: |
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Transition Matrix |
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===================== |
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We build the transition matrix using numpy. As an alternative, the SparseMatrixBuilder can be used here:: |
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In this example, we build the transition matrix using a numpy array:: |
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>>> import numpy as np |
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>>> transitions = np.array([ |
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... [0, 0.5, 0.5, 200, 0, 0, 0, 0, 0, 0, 0, 0], |
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... [0, 0, 0, 0, 0.5, 200, 0, 0, 0, 0, 0, 0], |
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... [0, 0, 0, 0, 0.5, 0, 200, 0, 0, 0, 0, 0], |
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... [200, 0, 0, 0, 0, 0, 0.5, 0.5, 0, 0, 0, 0], |
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... [0, 0, 0, 0, 0, 0, 0, 0, 200, 0, 0, 0], |
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... [0, 0, 0, 1, 0, 0, 0, 0, 0.5, 0, 0, 0], |
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... [0, 0, 0, 0, 0, 0, 0, 0, 0, 0.5, 200, 0], |
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... [0, 200, 0, 0, 0, 0, 0, 0, 0, 0.5, 0, 0], |
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... [0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ,0, 0], |
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... [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 200], |
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... [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.5 ], |
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... [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], dtype='float64') |
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We do not specify the row_group_indices. Thus, the following function call returns a sparse matrix with default row groups.:: |
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... [0, 1.5, 0, 0], |
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... [3, 0, 1.5, 0], |
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... [0, 3, 0, 1.5], |
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... [0, 0, 3, 0], ], dtype='float64') |
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The following function call returns a sparse matrix with default row groups:: |
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>>> transition_matrix = stormpy.build_sparse_matrix(transitions) |
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>>> print(transition_matrix)< |
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0 1 2 3 |
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---- group 0/3 ---- |
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0 ( 0 1.5 0 0 ) 0 |
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---- group 1/3 ---- |
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1 ( 3 0 1.5 0 ) 1 |
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---- group 2/3 ---- |
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2 ( 0 3 0 1.5 ) 2 |
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---- group 3/3 ---- |
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3 ( 0 0 3 0 ) 3 |
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0 1 2 3 |
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Labeling |
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================ |
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Next, we create the state labeling:: |
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The state labeling is created analogously to the previous example:: |
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>>> state_labeling = stormpy.storage.StateLabeling(12) |
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>>> state_labels = {'init', 'deadlock', 'target'} |
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>>> state_labeling = stormpy.storage.StateLabeling(4) |
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>>> state_labels = {'empty', 'init', 'deadlock', 'full'} |
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>>> for label in state_labels: |
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... state_labeling.add_label(label) |
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>>> state_labeling.add_label_to_state('init', 0) |
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>>> state_labeling.set_states('target', stormpy.BitVector(12, [5, 8])) |
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For the choice labeling this works similar:: |
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>>> nr_choices = 12 |
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>>> choice_labeling = stormpy.storage.ChoiceLabeling(nr_choices) |
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>>> choice_labels = {'loop1a', 'loop1b', 'serve1', 'loop2a', 'loop2b', 'serve2'} |
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>>> for label in choice_labels: |
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... choice_labeling.add_label(label) |
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To set the same label for multiple choices, we can use a BitVector:: |
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>>> choice_labeling.set_choices('loop1a', stormpy.BitVector(nr_choices, [0, 2])) |
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>>> choice_labeling.set_choices('loop1b', stormpy.BitVector(nr_choices, [1, 4])) |
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>>> choice_labeling.set_choices('serve1', stormpy.BitVector(nr_choices, [5, 8])) |
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>>> choice_labeling.set_choices('loop2a', stormpy.BitVector(nr_choices, [3, 7])) |
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>>> choice_labeling.set_choices('loop2b', stormpy.BitVector(nr_choices, [6, 9])) |
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>>> choice_labeling.set_choices('serve2', stormpy.BitVector(nr_choices, [10, 11])) |
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Reward models |
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================== |
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Now, we create the reward models, beginning with the reward model named 'served' which has state-action rewards:: |
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>>> reward_models = {} |
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>>> action_reward = [0.0, 0.0, 0.0, 0.0, 0.0, 2/3, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0] |
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>>> reward_models['served'] = stormpy.SparseRewardModel(optional_state_action_reward_vector = action_reward) |
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In the same way we can create a second reward model. This time we consider state rewards:: |
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>>> state_reward = [0.0, 1.0, 0.0, 0.0, 1.0, 0.0, 0.0, 1.0, 0.0, 1.0, 0.0, 1.0] |
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>>> reward_models['waiting'] = stormpy.SparseRewardModel(optional_state_reward_vector = state_reward) |
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>>> state_labeling.add_label_to_state('empty', 0) |
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>>> state_labeling.add_label_to_state('full', 3) |
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Exit Rates |
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==================== |
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Lastly, we equip every state with an exit rate:: |
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>>> exit_rates = [201.0, 200.5, 200.5, 201.0, 200.0, 1.5, 200.5, 200.5, 1.0, 200.0, 1.5, 1.0] |
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>>> exit_rates = [1.5, 4.5, 4.5, 3.0] |
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Building the Model |
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==================== |
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@ -93,9 +68,8 @@ Building the Model |
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Now, we can collect all components, including the choice labeling and the exit rates. |
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To let the transition values be interpreted as rates we set rate_transitions to True:: |
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>>> components = stormpy.SparseModelComponents(transition_matrix=transition_matrix, state_labeling=state_labeling, reward_models=reward_models, rate_transitions = True) |
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>>> components.choice_labeling = choice_labeling |
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>>> components.exit_rates = exit_rates |
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components = stormpy.SparseModelComponents(transition_matrix=transition_matrix, state_labeling=state_labeling, rate_transitions=True) |
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components.exit_rates = exit_rates |
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And finally, we can build the model:: |
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@ -103,20 +77,15 @@ And finally, we can build the model:: |
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>>> print(ctmc) |
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-------------------------------------------------------------- |
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Model type: CTMC (sparse) |
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States: 12 |
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Transitions: 22 |
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Reward Models: waiting, served |
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State Labels: 3 labels |
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States: 4 |
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Transitions: 6 |
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Reward Models: none |
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State Labels: 4 labels |
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* init -> 1 item(s) |
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* empty -> 1 item(s) |
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* deadlock -> 0 item(s) |
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* target -> 2 item(s) |
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Choice Labels: 6 labels |
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* loop2a -> 2 item(s) |
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* serve1 -> 2 item(s) |
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* serve2 -> 2 item(s) |
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* loop2b -> 2 item(s) |
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* loop1b -> 2 item(s) |
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* loop1a -> 2 item(s) |
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* full -> 1 item(s) |
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Choice Labels: none |
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-------------------------------------------------------------- |
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hannah
5 years ago
Matthias Volk